Sqlite implements all of the features you would expect from a relational database, including indexes, transactions, write ahead logging, consistency, and provides a SQL wrapper on top of its file API. _This is what makes it a database and not just a file format_. To the client, whether or not the API is happening over a socket or a locally linked library is irrelevant. How connection pooling works is also not relevant to whether it is a database or just a file format.

Calling Sqlite just a file format is kind of like calling Python "just a syntax specification." That's part of it, but we're talking about the actual implementation of it (probably CPython).