This is a regular pronunciation of the letter "E" in Present-Day English
Pronouncing words spelled with the letter "e" with the sound /i/ (the sound in the word fleece, also transcribed /iː/, especially in British English) is an English thing. This ultimate origin of this convention is the Great Vowel Shift; however, aside from words that were directly affected by that sound change, there are many words spelled with "e" that entered English later, where the pronunciation with /iː/ must be due to analogy with the spelling-pronunciation patterns of other words.
One or the other of these two reasons applies to many words of Latin origin.
The specific history of (semi)breve
The pronunciation of semibreve in English probably comes from combining the pronunciation of semi- and the word breve, brief.
Here is the Oxford English Dictionary's earliest citation:
1594 R. Barnfield Shepheard Content iii. sig. Eij No Briefes nor Semi-Briefes are in my Songs.
Brief, breve is taken partially from French, and partially from Latin, but it seems like the immediate source is mainly French.
The Latin word is brĕvis, pronounced in Classical Latin with a short vowel /ɛ/ in the first syllable. This form was not a direct influence on English.
In Old French, the following two variants seem to be found: bref and brief. I'm not sure what the full story is there, but the Oxford English Dictionary indicates that the word passed from French into Middle English in the forms bref and breve, with a number of alternative spellings, including some spellings with ee that indicate a "long" value for the vowel.
The spelling of brief with ie seems to have been introduced to English later.*
So I believe that breve is a case where the letter "e" represents a vowel that underwent the Great Vowel Shift. Both Middle English /eː/ and /ɛː/ shifted to present-day English /iː/; I'm not entirely sure which vowel this word would have had before the shift, as either could be written with the letter E.
It's not exactly pronounced as if it were a Latin word, but it's a bit similar
In the "traditional English pronunciation of Latin", which is more or less a French-influenced spelling pronunciation of Latin words that originated early enough to be affected by the Great Vowel Shift, the Latin word brevis is pronounced /ˈbrivɪs/, while the neuter form breve is pronounced */ˈbrivi/.
Because the ĕ in brĕvis is short, a purely Latin form semibrevis would be stressed on the third-to-last syllable according to Classical Latin stress rules, yielding a pronunciation very different from the English pronunciation of semibreve. However, semibrevis seems to only exist in Latin post-classically. According to references I can see online, semibrevis occurs in the Latin text Ars cantus mensurabilis, by Franco of Cologne, from the 12th or 13th century, so the Latin word does appear to be attested before the OED's earliest citation for semibreve/semi-brief in English.
I doubt that the English pronunciation of Latin was a direct source of the pronunciation of semibreve, but it might have influenced it somewhat.
*"The influence of Latin and French on the use and phonological development of the digraphs <ie>, <ea> and <ou> in English between 900 and 1800 CE", by Veerle Kruitbosch, page 15
