Likely not a bug, see edit below.
When using the MusicBrainz Tag Sources option, after selecting a result and clicking Next, the list of IDs shown on the left panel are all Recording IDs not Track IDs as the column header states (bug1). I'm not sure which ID type is meant to be displayed next to each song on the left, but for certain, those IDs aren't Track IDs.
When saving these suggested tags, these Recording IDs are mistakenly being written to MUSICBRAINZ_TRACKID instead of MUSICBRAINZ_RECORDINGID (bug2).
When viewing a release on MusicBrainz (e.g. this), the Track ID can be found by clicking the track number, and the Recording ID can be found by clicking on the song title.
Hopefully I'm not misunderstanding things here,
Version: 2.91 stable (current dev build doesn't list any relevant changes)
So after some more investigation, bug1 is not a bug. Most of the confusion I'm having here is is that Mp3Tag and Picard call these fields different things on the surface but it looks like they're handled correctly under the hood. The lesson here seems to be to not enter MusicBrainz ID manually and trust the software is doing it correctly because the fields are being displayed as one thing and being written as another. Examples (assuming all mentioned tags are prefixed with MUSICBRAINZ_):
On the MusicBrainz site, recordings (i.e. the hyperlinked titles on a release), are referred to as TRACKIDs in Mp3Tag but as RECORDINGIDs in Picard, but both actually write to TRACKID. Again on the site, tracks (i.e. the hyperlinked track numbers on a release), are referred to as RELEASETRACKIDs in Mp3Tag but as TRACKID in Picard, but both actually write to RELEASETRACKID.
I wish tagging was more standardised and transparent.to avoid such confusions.