I guess I would need some clarification regarding your use-case and setup. If you're referring to the computer name, that part is just for the display name purposes, it's not a "functional" part of the path.

Here's an example:

Here you can see how it looks when simply opening the drive letter with your mapped drive

And here's how it looks when you copy that same path as text:

So in your case the path to the files will always be the same as long as the drive letter stays the same.

I wish it were that simple. The basic use of the file explorer works as you have said.
Unfortunately it seems that there is a difference in the actual file location based on the user. I am using the exact same access keys and secret keys. But the users are creating something strange, the root folder seems to be different, and this comes down to Cloudberry, and how it handles the mounting of the S3 drive (I think).

More specific use:
We are trying to sync Revit files. The Central Revit file exists on the server (s3) and then makes a local copy of each file once a user opens it. Then Revit records the file location (the file on the server), so that when a local user saves, the file syncs and updates the central model. This allows multiple users to work on the save file (but not save at the same time - a busy error is given if both access the file simultaneously).

Anyway, here is a screenshot of the path from the Central file, when the error occurs.

The root folder can be set up when creating/editing your mapped drives. As for the issue with naming, that appears to be specific to the 3rd-party software you're using to synchronize your files. From what I see you're specifying UNC path instead of the disk letter.

Precisely.
The UNC path is being used.

Is there a way to make the UNC paths match each other? On our local system, the root of the server or .local is used at the beginning of the UNC path. Cloudberry prevents us from using a universal location? Is there a way to change the UNC root in cloudberry?

That is more of a Windows problem/limitation. If you map any drive using native Windows functionality it will behave in exactly the same way as within the Drive software.
I've also consulted with our Drive PM, and If your software is not able recognize mapped drives and converts the path to UNC automatically we can't really help with anything in that regard.

Thank you. I will look into this further.