The size of the key space in Bitcoin is just shy of 2256. There are a few different output types with different address schemes, but for most of these, the address space is a lot smaller than the key space. Let's assume you're inquiring about P2WPKH. There are about 2160 unique addresses for P2WPKH outputs, so about 296 keys map to each address.
You are therefore correct that hypothetically two users could each discover a different private key that map to the same address. If that were the case:
- There is no registration of addresses or other mitigation for key collisions. Simply, either user could spend funds received to the address.
- From a protocol perspective, there would not be a problem at all, as a transaction signed with either key would be valid.
- If they both notice in time, the users might get into a bidding war, creating conflicting transactions with higher and higher fees to claim the funds.
- If transactions from both users surface, other people might notice and be concerned about the key collision. Most likely this would lead to a loss of confidence in the method the corresponding keys were generated, but some users could lose confidence in Bitcoin instead due to misunderstanding the cause of the situation.
2160 is an extremely large number. As long as key generation uses reasonable entropy sources, it's unlikely that we'll ever see any two addresses collide no matter how long Bitcoin will get used.