How do you tag a mortgage lender that is not a bank?
The tag “money_lender” doesn’t seem to fit because that tag is associated with payday loan businesses.
I think you mean a mortgage lender’s agent, not a the actual lender. The actual lender will be in some large, head office, building, somewhere.
office=financial might be a starting point.
At least in the UK such places would either offer other personal finance services, or other real estate services.
shop=mortgage has some usage at taginfo: https://taginfo.openstreetmap.org/search?q=shop%3Dmortgage . It maybe that one of the more popular other shop tags is more appropriate; take your pick: https://taginfo.openstreetmap.org/keys/shop#values .
When I was looking at the list of what to render in that line I found the list at line 2664 in https://github.com/SomeoneElseOSM/SomeoneElse-style/blob/master/style.lua#2664 .