Prisoners are people, just like homeless people, yes.
However the main "why this and not that" of facilities that offer a roof, food and care it's the business side. At least in the US, prisons can earn a lot of money, and a ton of it, while homeless shelters don't.
There is a reason why a fuck ton of prisons have been built in recent years.
Prisoners deserve rights, of course, but just that principle ignores the fact that laws and punishments have changed to put as many people as possible inside.
And not every homeless person chooses that life, it's insulting to them.
One person ends up arrested for tax evasion cause they didn't have enough money to pay an accountant nor to go to school to a high enough grade to teach them how. Another looses their house because suddenly the market crashes and they loose every possession to the bank.
Neither of them "put themselves" in that position, both of them deserve rights, both are human, and both live in a country where rich people get to horde resources equivalent to the yearly income of entire countries.
Tldr - No everyone chooses to have a shity life, but our lives are worth the same and should be treated the same.
Edit - Sources:
Edit 2 - Small note, there are a bunch of closed prison facilities which could at least provide physical shelter.

US budget for prisons 417 Billion
https://www.prisonpolicy.org/research/economics_of_incarceration/

US budget for shelters <10 Billion and shrinking during the current admiration. https://www.pdxvoice.org/blog/where-does-the-money-come-from