No real sync exists like you mention exists.
The closest solution is what I and a few others do. Using a cloud replication service, such as Micorosft OneDrive, Google Drive, Dropbox....
You put the catalog, all pictures, and make sure to include preferences and plugins in the same base folder, in the replication service. Now, everything will be replicated between the computers. However, and this is a big one, you CANNOT open Lr on the second computer until replication has finished between both computers. Otherwise, your catalog can become corrupted and/or end up with duplicate conflicts.
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