Picard supports reading existing fingerprints from the acoustid_fingerprint tag, but it has no option to save calculated fingerprints there.
While usually this was considered to be not needed, as the fingerprint can always be calculated again from the audio and can add rather long data to the file, there are some use cases and some users using this. E.g. one user on the forums described a workflow where they add this tag directly after ripping to avoid having to redo the calculation later on.
Add an option for this, but keep it disabled by default.