Take a bag which can cover the wheels, either one big one or two smaller ones and you can carry it into your hotel room in almost all hotels.
If you use them in places where they get extremely dirty, you can bring a brush or ask the hotel for a rag and some water to clean them up a bit.
I expect many hotels will be happy to have you leave it somewhere like the hotel's garage, but it will be hard to lock an item as small as this in such a way a thief will find it hard to take it out.
And luggage rooms are meant for luggage, and some will get pretty much filled to the brim at times and an odd shaped item may get in the way.
Charging will likely not be a problem in modern hotels. In older ones, you may have to check with the hotel that the wiring is not going to be a risk.
I do have experience with (recumbent) bikes and trikes in hotels, most are very happy to help you. But there is no standard solution. My bikes and trikes will not fit in the average hotel room, your scooter will.
I have used a human powered scooter in about this size and I know it is not hard to store them in a smallish room.