SQLite is not "just a file format" anymore than say MS SQL Server is a file format. SQLite is a RDBMS in the form of a library and uses a particular file format for persistent storage. But it can also run purely in-memory: https://www.sqlite.org/inmemorydb.html without any file at all.
Being a RDBMS is not defined by whether the engine runs in-process or as a server in its own process.
SQLite is just a file format. If you want it served up over some kind of server, you have to build your own (and most people do), or use a server that somebody else has built for you (there's a couple out there).