Not necessarily.
Ever heard of linkable systems? They can detect when multiple proofs come from the same person, even if they can't identify who that person is. The system can also force reuse of the same secret, which stops the "infinite proof factory" problem.
Unique secrets can also be tied directly to identity. For example, if the ZKP is about knowledge of a secret key bound to your identity, then you can't just mint 5000 independent proofs unless you also have 5000 identities.
There's also the concept of nullifiers, used in privacy-preserving identity protocols. A nullifier is basically a one-time marker derived from your identity secret that prevents double-use of a proof.
On top of that, zk-SNARK-based credentials or verifiable credentials can prove "I am a unique registered person" without revealing which one. These systems enforce uniqueness at registration, so you can't magically spawn 5000 ZKPs that all look like 5000 humans. Similar ideas exist with linkable ring signatures and even biometric-based ZK proofs.
So there are plenty of ways to counteract your "5000 ZKPs per human" story (what's usually called a Sybil attack).
If you're being pedantic, yes: a bare ZKP alone doesn't enforce "one proof = one person", but ZKP + uniqueness enforcement (nullifiers, credentials, commitments, etc.) does, and that's what I had in mind. I thought it was obvious, but then again, nothing is obvious, and I should have specified. My bad.
In any case, people ought to know just how powerful and useful these ZKP-based systems can be when designed properly. I think this is the only way forward if we want to preserve our privacy, and at the same time we want to prove we're human without sacrificing anonymity, or verify we know the password without revealing it, or prove we're eligible to vote without revealing our identity, or demonstrate we meet age requirements without showing our birthdate, or verify we have sufficient funds without disclosing our balance, or show we're authorized to access something without revealing our credentials, or verify our qualifications without exposing personal details, and so on.
Edit: excuse the technical brain dump, I literally just woke up. I hope this helps to clear up some things, however.
Happy to dig deeper if you want.