I stumbled about this problem somewhat often in the recent past and noticed it for more tags than just this one but (stupid me) can't remember what else tags were affected. Is this a known problem? Are all tags "multi-value capable" by default or is supported added for each specifically?
For now I can report that MusicBrainz Picard writes two MUSICBRAINZ_ARTISTID tags but mp3tag can only read one (and if I re-write the tag only one tag is written back, thus I lose the other tag).
Then you would have to show us how you and where treat this tag field.
If you modify a value for 1 instance of this field in the extended tags dialogue, then the other tag fields of the same type are preserved.
If you have a display like artist1\\artist2 and you overwrite that to artist3 then the former data will be lost.
I can not confirm this behaviour:
As you see, I can write 3 different values in the same MUSICBRAINZ_ARTISTID in Mp3tag. They will be shown as 123\\456\\789 in the Mp3tag tag panel.
I have a MP4 with all tags removed. I tag that MP4 with MusicBrainz Picard which writes two MUSICBRAINZ_ARTISTID tags. I read that MP4 with mp3tag but I only see a single MUSICBRAINZ_ARTISTID tag. I read that MP4 with MusicBrainz Picard again and still can see both MUSICBRAINZ_ARTISTID tags.
EDIT: Writing two MUSICBRAINZ_ARTISTID tags with mp3tag is possible, and those two tags can be read by mp3tag too. MusicBrainz Picard will also read those two tags. But two tags written by MusicBrainz Picard can't be read with mp3tag.
I tagged a file with MB Picard and a different file with mp3tag.
MediaInfo can see two tags in both files
foobar can see two tags in the mp3tag one but only one tag in the Picard one.
I'm not familiar with MP4 and can't say for sure, which version is closer to the expected standard definition. Maybe someone with more experience about MP4 can tell you.
Note:
The numbers itself (987/789) are different for better visible distinctness where they come from.
Mp3tag doesn't currently support the format in which MusicBrainz Picard is writing multi-value fields. I've made a note regarding this issue and will look into supporting this.
There is some controversy around how to do multi value tags with MP4. This is not really specified, and hence software took different approaches. See the following Picard ticket: