Historically, calibre used to use sqlite3, but I switched to apsw for calibre 1.0 (part of the big database layer rewrite) because it has better performance, and more control over sqlite semantics (apsw is sqlite specific whereas sqlite3 has to conform to the more general DB-API).
Not to mention that one is no longer tied to whatever sqlite is shipped with python.
However, as for your original issue -- I am not aware of any incompatibilities between 3.6 ans 3.7 as long as the db is not using any features that were introduced in 3.7
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