it's more of a Bm add4 (or add11).
with thumb to mute:
with thumb to fret:
and, of course, that can be tactically muted/strummed to mute the e at the top, making it a straight Bm too and giving a little melodic interplay between the d (7th fret g string) and e (open e string) at the top of the chord. Their spacing apart (separated by the open b) makes it not too tricky to target these notes and highlight them while strumming, which is what your man does here.
To mute the ringing e string entirely for a straight Bm chord, you can let the inside of your left hand fingers touch the bottom of the guitar neck. You don't have to, but if you want to it's a good tip for how.
Abcjs transcription of the 4 possibilities just for fun:
%%annotationfont Times-Italic *
|"^no thumb" "_Bm add4/F#"[FBBde]| "^thumb""_Bm add4"[B,FBBde]|"^muted top w/o thumb""_Bm/F#"[FBBd]| "_Bm""^muted top w/ thumb" [B,FBBd]|
the first is the one most employed here