If you think of it strictly from a reward standpoint, every hotel group encourages this behavior. Most elite levels come with far less stays than nights, so if you are going for elite status switching hotels is an easy way to get there faster. A lot of bonuses, elite welcome points or promos, are given per stay. So if you switch hotels every night you get a ton more points.

If people switching a ton was an actual problem, I'm sure that rewards programs would quickly adjust their terms. Clearly there are not enough people switching hotels every night to cause an issue. People are generally pretty lazy and don't value the extras enough to switch hotels in the same city. Are you going to ask the hotel to store all your stuff every day while you go out, come back to get your stuff and then go to the next hotel? Also people often want to just be in one part of town, so moving makes their activities more inconvenient.

If you don't want to move every night, move around every other night. You'll get three stays without too much headache and be that much closer to the passport bonus too.