From the Collaborative Bond and Money Market Data Portal
"Monetary financial institutions" (MFIs) are resident credit institutions as defined in European Union (EU) law, and other resident financial institutions whose business is to receive deposits and/or close substitutes for deposits from entities other than MFIs and, for their own account (at least in economic terms), to grant credits and/or make investments in securities.
More precisely, Regulation ECB/2013/33 concerning the balance sheet of the monetary financial institutions sector (recast) defines MFIs as resident undertakings that belong to any of the following sectors:
In accordance with Regulation ECB/2013/33, the ECB maintains and regularly updates a list of MFIs. (For the complete legal framework, see here.)
Within the universe of Financial Institution Groups (FIGs) Branches typically have different Monetary Financial Institution Codes (MFIs).