Before anyone says "oh well maybe there's a genuine technical reason", it didn't have this restriction for years, and then suddenly started doing this. Also, the app communicates to Fido servers via an online API, and you're logged in to your account, whether you're on one kind of network or the other should be irrelevant because everything should be happening server side anyway.
Also, it's not just "you can't be connected to Wi-Fi", it's you can't have Wi-Fi enabled period. Even when you're not connected to a network and it's already directly connected to data, it doesn't work.
Also also, if you run of regular data but still have unlimited sessions left, you can't actually use them without first buying a data add-on, since you can't start your session on Wi-Fi and then switch to data.