It has been confirmed that The Crew does have a hidden offline play capability:

I can now confirm that local save data functionality is indeed built into all our copies and is not absent from the game, we have even obtained a sample save data file. This + the fact that we literally see offline mode in action in the prologue means that there 100% is an offline mode, if anyone was doubting at this point.

Save data dumper has been released to allow dumping your save data that can be used to play offline with a mod later:

Yes. a server emulator for offline play is in development by r00t0.

The save file dumped from ubisoft servers and is planned to be used for the offline mod so no progress will be lost.

However, you only have a limited time, after 31 march the servers will go down and your save file will be forever lost.