If an ODBC linked table was created with a File DSN, then it's not using any DSN.
MS Access will pull the information from the File DSN into the linked table and build a connection string using that information directly, rather than storing the path to the File DSN. Once this connection string is built, the linked table no longer references the File DSN for anything. For better or worse this functions exactly as if you had hardcoded the connection string and never used a DSN in the first place.
On the other hand, creating a linked table with a machine data source behaves as expected (or at least, as I would expect). The linked table maintains a reference to the DSN it was created with, and will only work if that DSN is present.
To see this for yourself, the connection string can be viewed by hovering over the linked table:
- If the connection string was created with a Machine DSN, it will have a DSN property: