"Philadelphia is my city."
Translation:Philadelphia est urbs mea.
Yes, the genitive is generally used to denote possession (such as in “the king’s throne). The genitive can also have other uses depending on context (which as far as Duolingo goes I risk giving obsolete information here, but they should be learned if you want to get a better understanding of the grammar): 1. Possessive genitive 2. Partitive genitive (part of a whole, as in “pars hominum” = “a part of the men”) 3. Objective genitive (as an object of the action, as in “Metus hostium” = “a fear of the enemy”... There are many more uses than that, but these are the ones I run into most frequently when translating the more ubiquitous writings.
Possibly beyond the scope of duolingo, but I know you would say: Mihi nomen est (blank) for "my name is (blank)". Would it also be appropriate to say "Mihi urbs est Philadelphia" to say "My city is Philadelphia" (or Philadelphia is my city, as in the prompt). I know that the dative of possession isn't often taught at the beginning of learning Latin, and this sentence seems to come up early in duolingo, but I used this construction in the placement quiz and it wasn't accepted. Thoughts on whether it would be an appropriate translation or not? Something I have yet to learn while teaching myself Latin is when the dative of possession is/is not appropriate, so I'm interested to know if there is a particular reason why it wouldn't be appropriate here.