I am from Austria and would like to visit the UK with my own car. The plan is to travel through Germany and France to take a ferry to England (Calais-Dover), stay 1,5 weeks in the UK and then go back the same way. I plan to stay mostly in bigger cities like London, Manchester and York etc, but also some smaller places like Portsmouth. Is it safe to drive and especially park a foreign, middle-prized car (just a VW Passat Variant from 2016)? I have heard from some people that it is less safe now to drive in UK with foreign car registration plates since xenophobia is on the rise especially in England. Also, do I need to have anything special in my car, when driving to the UK?
Foreign number plates
I think generally you won't have a problem having a car with foreign number plates. While in this post-Brexit-vote world there certainly is increased publicly-expressed xenophobia I don't think it's to the level you worry it is.
Driving in cities
If you intend to drive in central London (something I would advise against, personally), you should look into the congestion charge and how you intend to pay it. If you drive through the zone and don't pay you could be subject to a fine. But my recommendation is to use public transport to get around within central London; the tube if you want speed, and buses if you want to see the scenery go by — or just walk! You can find somewhere to park in Greater London and catch the tube or bus into central London. You'll find the experience much more pleasant than driving, I'll bet, and you won't have to worry about finding a place to park. (That page also mentions the Ultra Low Emission Zone — your car being quite new I don't think you have to worry about paying that, but you will need to register your vehicle so they know you don't have to pay it.)
In general driving into city centres, especially in older historic cities, can be slow, and parking can be hard to find and expensive. For each city you plan to visit, I would suggest looking up parking, traffic, and any park and ride schemes (or places with good public transport links and large carparks nearby, essentially as "unofficial" park and ride).
Requirements for driving
If you're driving through France do note that they have some regulations on things you must carry in your car (a warning triangle, a high-vis jacket for each passenger, and a couple of breathalysers, as well as a pollution sticker if you're driving in Paris). The UK by and large doesn't have such regulations, but if you're driving through France you'll need to follow them!
Since you'll be driving on the other side of the road in the UK, you should get headlamp deflectors to avoid dazzling other drivers (your headlamps will be designed to point slightly to the right if your car is built for driving on the right hand side of the road; when driving on the left this will dazzle other drivers). I don't know if those are a legal requirement in the UK but they're certainly a courtesy.
Things to know about UK driving rules
Unlike some countries, the UK doesn't allow you to turn left (or right) on a red light. Red means stop, no exceptions. There are also no jaywalking laws in the UK so expect people to cross the road in front of you, especially in London. At older ("Pelican") pedestrian crossings, the light sometimes flashes amber — this means "go if nobody is on the crossing". Newer ("Puffin") crossings have sensors to detect people on the crossing so no longer display this indication. Zebra crossings (pedestrian crossings without traffic lights) are very strictly followed by road users in the UK — while technically you can drive if nobody is on the crossing, almost every road user will stop if it looks like someone wants to use the crossing.
In the UK, speed limits are expressed in miles per hour. If your car doesn't have MPH printed on the speedometer I would suggest learning the conversion for multiples of 10. If no speed limit is displayed, the national speed limits for ordinary cars are 30mph in a built-up area (if you can see streetlamps, it's probably a built-up area), 60mph on a single carriageway (ie country road), and 70mph on a dual carriageway or motorway.
Overtaking on single carriageways is difficult in a left hand drive car, and I wouldn't blame you for not attempting it. Legally you can do it anywhere where the single centre line is dashed, or, if there's two lines, where your side is dashed and the opposite side solid. But use common sense, as not all roads are marked up with the safe spots.
Overtaking on dual carriageways and motorways is a bit simpler. Always overtake on the right, don't overtake on the left (referred to as undertaking) unless you're stuck in slow traffic. For this reason, in the UK traffic should keep left unless overtaking or getting in lane for a junction. In practice this isn't always followed by everyone, but staying in the middle or right lane for too long even if going at the speed limit is frowned upon and, I believe, actually illegal as of a few years ago.
The UK famously has a lot of roundabouts. Make sure you're familiar with the use of roundabouts, especially in the UK. Give way to people already on the roundabout; don't stop on a roundabout to let other cars on (unless there are red traffic lights, emergency vehicles, or if not stopping would cause a crash of course!). When approaching a roundabout, indicate appropriately (though beware of others not doing so) — left if you intend to take the first exit, right if you intend to travel more than 180 degrees around the roundabout according to the sign (switching to a left indication just after you pass the last exit that isn't yours), and don't indicate if you're going between the second exit and the exit at 180 degrees inclusive (again, putting on a left indication as soon as you pass the last exit that isn't yours). If there are two lanes on approach to the roundabout, unless signs or road markings show otherwise, the left lane will be for all exits from the first to the exit at 180 degrees (as displayed on the sign), and the right hand lane will be for exits further than 180 degrees round the roundabout. A mini roundabout works pretty much the same as a normal roundabout from a car's perspective, but just be more wary of giving way as the roundabout is much smaller, and expect longer vehicles to drive over the central dot!
While technically not allowed by the Highway Code, it is very very common for people to park against the flow of traffic (on the "wrong" side of the road) in on-street parking (as found in residential areas). Don't be afraid of doing it yourself if you would otherwise lose out on a good parking space, and don't be alarmed if other cars do it in front of you. When parking on a street, always look out for signs advising of parking restrictions, as they are many and varied. Some car parks are "pay and display" so you must buy a ticket from a machine right after you park and display it in your windscreen; have a good quantity of change ready as unless you want to spend ages signing up to some mobile app or service, many of these places only take coins and don't give change. Other car parks give you a ticket when you enter and require you to pay off the charge in a machine just before you leave.
My suggested alternative to driving
Personally I would recommend getting the train, as I find it a much more pleasant way to get around the UK (our motorway traffic jams can be legendary), but that's just me :). Manchester, York, and Portsmouth are all well-connected to London by rail (and Manchester and York are well-connected to each other), and getting there by train will almost certainly be quicker than by road, and usually bring you to a more central location than you could ever reasonably expect to park. While trains are expensive if you want maximum flexibility, if you're willing to give up flexibility and commit to a specific train in advance, you can find tickets for very cheap. And on the other hand as a European citizen you can buy an Interrail one country pass for the UK (the UK is very good for Interrail, as only the overnight sleeper trains have mandatory seat reservations, and optional reservations can be obtained for free), or a Global Pass for every participating country in Europe except your own.
For that short time period you dont need to do anything to your car and it will be perfectly safe, we drove our car from Stirling (Scotland) down through the UK and over to France and back up to northern Sweden, its a beautiful drive and we had no real challenges.
One thing we did do however was to buy a type of plastic film to put on the front head lights since the UK drives on the left you might find that your head lights are angled slightly and it did annoy me but we just asked at a gas station and they had a film that you put on and then tore off when we left the UK.
A also it takes a bit getting used to driving on the left, specially when you want to over take on single lane roads you will find that visibility is not the best but just be careful and you will be totally fine.
Disclaimer: I have never visited the UK but being an avid reader of the https://www.theguardian.com/uk/money page, I do like to warn you about some possible issues that can arise especially to a tourist/foreigner in urban areas of England.
My answer mainly deals with issues like Parking/Congestion charges.
As per my understanding a significant number of both public and private institutions have outsourced the handling of their customer parking areas to private firms. These private firms enforce strictly and fines easily exceed GBP 100.
In some public parks, you need to enter the number plate of the car completely on the parking meter before you can print it out. These generally are tailored to GB number plates and hence sometimes do not support foreign plate format. Also accidental switching 1 and I, 0 and O make it invalid although you have paid.
When in a shared car park between multiple enterprise (eg: McD, Starbucks) parking the car in a wrong section also leads to huge fines.
Also watch out for the time limit on various free parking areas especially at gas stations/supermarket. Exceeding the time is also strictly enforced.
Congestion charge needs to be paid online and at least several people have complained that payment gets taken and refunded later and then they fine you for not paying.
In my experience, none of these could ever be an issue for me, when driving across Germany, Italy, Spain, France, Austria, Switzerland... So from my perspective these seem to be a big deal specific to the UK.
That said if you are certainly visiting the rural parts of Wales or Scotland, I would still recommend taking a car (rental or yours).
note: If any locals would like to correct any mistakes/misunderstandings in this answer, kindly do so.