This is essentially the same issue discussed here: How to Edit Final Cut Project on Two Macs… - Apple Community
While it's understandable that users would want to use cloud services to replicate an active FCP library, the root problem is MacOS does not have "mandatory" file locking. It's not like Windows whereby a file opened for writing automatically blocks readers. That is why on MacOS you almost never see an error like "The process cannot access the file because it is being used by another process."
You can use Finder to copy an entire library even though it's open and in use by FCP. But you aren't guaranteed to get a consistent version of the library. It could easily be damaged if it's copied during FCP data writes.
Likewise the Dropbox sync program can easily read files from an FCP library that are being actively updated. Those updates will be transmitted to other users who have access to that library. When the library is read, it could easily be halfway through a transaction involving multiple SQL tables. Those SQL tables must be updated as a single "atomic" operation, else the data will become inconsistent.
On both people's machines, it's a local on-disk library -- but one in which Dropbox is constantly monitoring it, reading any changes and propagating that to the other user's version of that library.
This is inherently risky but (depending on the circumstances) you may get away with it for a long time before you encounter a damaged library and lost edits. If neither source machine never hangs or crashes, and if the recipient is not running FCP when the sync takes place, and if there's never a network glitch or error during the sync, it will probably work. But if any of those things happen, it can damage the library.
Imagine a bank database where debits and credits between different accounts are happening. If you just grabbed a snapshot of that, it would contain debits with no matching credits -- IOW money would be missing. Transactional integrity would be lost.
I believe that is likely why FCP will not open a library which is stored on Dropbox if using the updated Dropbox file provider interface. This closes a risk which should have never been allowed in the first place. It is very likely not a bug, it's by design.
It is possible to transfer FCP libraries using Dropbox but those should be offline and ideally ZIP'd or compressed using CRC checksum protection.