Annoyed already pointed out some good thoughts. Some more from my side on top of that.
What you have to consider is that the distances in Western Europe are larger (Spain, France, Germany and Italy are rather large countries) but the distances in Eastern Europe can be more troublesome despite being smaller (quality of infrastructure, visa requirements, Cyrillic font, lower quality of GPS maps etc).
The question is really if you want to do a "As many countries in as little time possible" or a "I want to see something significant every country, and do not care about time", or anything in between. Each time, as you can imagine, your trip will look vastly different. I can only tell you that there is so much to see along the way, that the faster you go, the more you waste money and time in the process. There might be more value to see Rome AND Paris in the same week for some people than to see only one of them for a week, but to see 5 countries in a week clearly is leaving the travel aspect and moving more into racing, where the completion of the trip is the accomplishment and not what you have seen. Everyone has to decide for themselves.
There is (as far as I know), no 'famous' challenge like this in Europe. The issue is a bit that the landscape in Europe is changing faster than in the USA, with countries splitting and re-uniting, with the EU expanding and with the economic changes in Europe there are always countries who claim to be a part of Europe despite many people saying that they are not, and others who try to distance themselves while most people would consider them a part of Europe.
So let's assume that you just want to see the biggest capitals in Europe with a bit of flexibility (such as Hamburg instead of Berlin etc):
- For the first 9 countries, you could do northern Europe and Skandinavia: London, France (Lille or Paris), Brussels, Amsterdam, Hamburg, Copenhagen, Malmö, Oslo, Stockholm, Helsinki (with a ferry). Car and or train exists for the whole route.
- Then, the next round, 9 more countries, you can go through Eastern Europe: Helsinki, Tallinn, Riga, Vilnius, Minsk, Warsaw, Prague, Vienna, Maribor, Zagreb
- The last large round of 9 more countries could be the southern part of Eastern Europe: Zagreb, Budapest, Kosice, Lviv, Chisinau, Bucharest, Sofia, Bitola, Tirana, Rome.
- The last batch of 3 countries to the east coast would be then: Rome, Geneva, Barcelona, Lisbon
While there are some really logical steps in there (Spain-Portugal) there are certainly others that are completely arbitrary an can be done in 50 different schedules. Countries can be changed, different cities chosen. It can be argued for example if Belarus is a country you want to visit or if Portugal is a good place to end the trip for example.
But good thing with traveling through those rather large cities is that train connections exist for most of these, and you can do most of the traveling without planning a long time ahead, since you will most likely be able to find a seat in a train and a hotel 2-3 days ahead of arrival. The biggest risk that you have is when you want to travel in very fast or special touristic trains in high season or rush hour or stay in business hotels when there are big fairs or conferences in a city.
You can rent a car, but be aware that the price including insurance can be quite hefty for such a long time. But there is definitely no issue picking up a car in one country and dropping it off in another as long as you rent with one of the large companies. And if you plan to visit places outside the large cities, a car very quickly becomes a necessity, since a lot of countries in Europe do not have an extremely good infrastructure when it comes to countryside locations.
As another alternative, you could also try to see a possibly large diversity in Europe. While how to achieve this is of course VERY subjective, one could do this by ethnicity. There are more than 60 according to Wikipedia and more than 30 with more than a million people, so I am sure you can chose many countries to visit, but would go only to one among Germany, Austria and Switzerland for example, but spend more time in the UK & Ireland since there are Scots, Welsh and Irish people.
So as you can see, it really comes down to what you want to achieve. If you narrow down what type of trip you want to have, I am sure you will have some more concrete questions and get more targeted answers.