@DavidCarlisle make4ht seems to try to prevent fontspec to use open type fonts (as the binary can't handle them,
tex.stackexchange.com/a/122585/2388). Its patches in
usepackage-fontspec.4ht
contains
\cs_set:Nn \fontspec_set_family:Nnn{\def#1{\relax}}
, and this then fails when unicode-math tries to setup a font by using the family name stored in
#1
. (there could be more problematic \relax around)