Generally yes. Some Credit Unions may be willing to open an account without the same level of documentation, but opening a bank account can be a real hassle.
The first bank application I checked requires a "residence or street address" and specifically forbids PO boxes. You also need to have current, government-issued ID, which also requires proof of residence.
It's simply not easy at all to participate in the modern economy if you are a transient or anyone else without a permanent address.
A while back, I had a mailbox at one of those Mailboxes etc. businesses, and I was able to have all sorts of things that don't allow PO boxes sent to it, since it looks like an apartment address. Worth a try for anybody in that situation.
Using such an address as your "permanent residence" could very well constitute fraud. You certainly don't live in a Mailbox store. Just because you can receive mail there, doesn't mean it's your address.
I went through this hell. to get the PO box you need a residential address, you need ID with an address. I moved to a new state and was temporarily without residence or local ID, and as an experiment I tried to see how much I could accomplish without giving up an address and/or phone. It's impossible, or very nearly so.
I did find options though. One thing that might work is, find a Catholic church. Some of them will let you have your mail sent there for you if you're homeless. It didn't work for me because I knew I would be taking advantage of a good will service for people who are actually unprivileged.
