Chicago is grappling with local backlash against moving homeless migrants into a new government-run tent encampment as temperatures drop and more migrants continue to arrive.
It's correct how it is. It's acting as an adjective, as if you said "red shelter." "Migrant's shelter" would mean the shelter belongs to one migrant. "Migrants' shelter" would mean the shelter belongs to a group of migrants, which is also incorrect because they do not own it. Hope this helps :)
Honestly, I will say that I would consider “migrant shelter,” “migrants’ shelter,” and “migrants shelter” as all equally correct, with minor differences in shade of meaning.
I would not, and I'm not sure why. Maybe because I differentiate between living and nonliving things for some reason? I would say the lions' cage, for example. Actually, unless I was contrasting it, I would say "the lion cage", regardless of the number of lions it houses, but that might just be a fixed expression in English regarding zoos and such. I would not say dry goods' pantry or pants' drawer.
I find it fascinating how we internalize the rules of language in a way that sometimes leads us stranded to explain them.
I do have to say that I’m going to start using pants’ drawer, though. The way it stacks plural on top of plural and then shoves the possessive at the end is brilliant.
It's obviously not the shelter of the migrants, they don't own anything, they don't deserve to own anything but there are so many coming into our beloved country that's why it's plural!1!!1!! /s (I genuinely don't know the correct answer, English isn't my first language either, but I guess 'migrant shelter' would simply sound awkward and there is some truth to my bad right-winger xenophobe LARPing attempt, which is that the migrants sadly don't own the shelter so the apostrophe would be incorrect in either case)