IANAL: They are Payment Processors, technically serviced by banks, and not a Money Services Business. The difference between them and say AirBnBs transactional payment service is a technicality, but I'm not a lawyer.
1. They might actually know what they are doing or 2. they aren't doing payments of the sort that needs this sort of license.
Stripe is backed by Wells-Fargo, aren't they?
Stripe uses Wells Fargo as an underwriter for the Merchants it on-boards. Wells Fargo did not invest in their earliest rounds but may take an equity stake in later/future rounds.