The Fairphone 3 sports a capacitive fingerprint scanner as can be seen on this picture (the middle loose component):
I am not sure which exact model it is.
It (that is, the hashes based on the input data of your fingerprint) should be stored in the secure element of the ARM processor. This storage is done locally, not in the cloud. If it is, that is huge a security risk and design flaw. We know it is stored locally only though because the data is not available during an Android backup migration.
Having the information could be used to, in theory, make a hostile fingerprint reader which could be used to get access to the device and/or the encrypted storage.
In practice, this attack vectors is needlessly complex, and only interesting for advanced forensics by nation states. If you are within the scope of these, there are much easier made within your attack surface ie. you are worrying about the wrong things. For example, in The Netherlands you can be forced to use your finger as authentication because it is not something you know, but it is extremely easy (anecdotal) to figure out someone’s PIN by just looking over their shoulder. You should start with creating viable entropy with your PIN, including taking into account partial/full knowledge (your birthday is not a good PIN if your adversary knows OSINT).
Furthermore, what is important with authentication such as these, is the amount of false positives. FaceID and TouchID have a certain amount of (tested) false positive rates e.g. 1 in a million (just an example). We do not yet know the false positive rate of the Fairphone 3 fingerprint reader. I will do some preliminary testing as soon as I have the device.
If you are worried about your privacy: your fingerprint itself is only used to verify the hash. You’re leaving your fingerprint everywhere you travel, and it is relatively reasonably easy to construct a fake finger(print) which can be used to authenticate.