Not necessarily in this case. Firebase is a gateway to paid Firestore, Storage, and Google Cloud. They give away Auth for free and is ISO and SOC compliant. An argument can be made for their free analytics offering though.
To be clear, are you saying one shouldn't be concerned about running their webapp on Firebase/store because of the regulations surrounding what Google can do with that data? (customer usage changes for choosing when to start building competing products, for starters)