In general, would hotel staff find it unusual/weird for a customer to stay at a hotel for an extended period of time (e.g. 3 months)? Wouldn't that mean more money for them? Given the current crisis right now, would hotels prefer to have a single customer stay for a long time or a multiple customers stay for a short time?
Depends on the place, often, and what they're aiming for.
For example, cheap hostels often have a rule about locals not being permitted to stay more than say, 2 weeks (to prevent down and out people taking advantage of cheap accommodation, ruining the traveller 'vibe' that the hostel might be seeking).
Have also seen a limit similarly on actual travellers - they want travellers, not backpackers on their working holiday just using it as a base. This might be for the mood, or also because they might rely on commission from activities for example, that they get from frequent different travellers.
Saying that, a friend lived in a hostel in London for 18 months, while working. I've done a few weeks in a single hostel in Argentina, Vancouver, and Melbourne, Australia, and gotten to know the staff, shared BBQs, drinks and the like.
As Mark already mentioned it depends on the hotel/hostel type. He is giving one version. I've seen many places having such policy. But there is the opposite case. There are many examples of rich people who decided to live in a hotel at a given moment in their life. And that never was a problem for the hotel.
A known case is Calouste Gulbenkian. A rich Armenian who settled in Lisbon and lived 13 years in Hotel Aviz.
But there are many more examples. Including people who live in cruise ships most of the year (these are kind of floating hotels).
I work away from home and regularly stay at hotels for a month or more, sometimes going home for weekends and sometimes not. My longest stay was for 3 months late last year at a hotel near Amsterdam, Netherlands (Joy Hotel - basic and cheap, but very clean and nice).
Without exception, the hotels have been very welcoming and happy with my extended stays. I have never experienced anything except goodwill. It's nice to get to know the staff a little better, learn their names, and chat with them regularly.
I think most hotels are glad of both the income and the vote of confidence you're giving them by staying their regularly.
Multi-weeks and months is not unheard of in the hotel business; a lot of people stay in hotel, both for tourism or business for a long time.
I does not matter one way or the other; it might even be easier on the staff as they do not have to make the room every day.
I think that people staying one night in a hotel is more burdensome.
UNLESS the client does not allow hotel staff in the room for regular maintenance (cleaning, bed changing...) and it result in the room degrading. (I assume there are rules and regulations for that)