They could, sure, but many don't. I think you're fine. I have personally never seen this happen when I've renewed my leases; all one-time fees were upon signing the lease for the very FIRST time, first year, and nothing else. They raise the rent, maybe add more expenses, like pet rent, but no, never had to pay the pet fee over again.
So yes, legally they can charge you whatever they want if you renew, but I don't think they will charge the pet fee again.