It seems MB (web edit) does not support thin space (U+2009) and narrow no-break space (U+202F).
You can enter (or paste) it for track titles in the Track List tab, still have it in the Recordings tab but it’s substituted in the Edit Note tab by a regular space.
The narrow no-break space is recommended for some cases, e.g. grouping of digits (German) and would be fine to have.
Is there a special reason for not supporting it?
Or is there another reason for replacing it by a regular space?
I found Should non‐breaking spaces be used? If so, when?
It is also somehow related to spaces (also the related issues) but it’s not about the thin/narrow (non-breaking) space.
And now I found https://musicbrainz.org/doc/Internationalization#Unicode_issues (incl. Typographic spaces).
Following this: They should not be used. 
There’s a ticket I made for a similar issue.
I agree, space variants (different width, nonbreaking) should be supported.
For example, when an artist stylizes words with spaces as in “V O I D”, the spaces inside a word should definitely be non-breaking and I think the spaces inside a word should be thinner than the spaces between words (in this case, “V O I D I” looks very confusing, the last “I” is the roman numeral).
It’s also the character to use before French punctuation :;!?» and after « and also for grouping/separating number thousands.
I guess it would bring complexity, inconsistencies and ping pong edits. Like the apostrophe edits we have or, worse, the hyphen edits. 
I wanted this in the old forum or mailing list, but now not so sure. In which I said that it looked unusual and strange to the eye to see full spaces in French punctuation, when you are used to printed French.
These things, most people cannot see or type them.
Well, since MB has a rather high claim on correct data (incl. e.g. typographically-correct punctuation preferred) we have to live with some of such kind of edits.
Therefore it should IMHO also support as many characters as possible.
I don’t see a real technical reason for not supporting e.g. the thin spaces. For me they are a Unicode characters like any other.
To avoid ping-pong edits it would be fine to have a better support/check in the edit interface, e.g. a warning when removing typographical characters.
(Personally I’d like also better support at initial edits, e.g. checks, help for entering correct characters, etc.)