Bank sponsor is a less common word order for sponsor bank, the licensed bank that allows a fintech, payment company or other non-bank to reach the financial system through the bank's own licences and memberships. That entry covers the model in detail.
A sponsor bank might hold card scheme membership so that a program manager can issue cards under its BIN, act as the originating bank for a fintech's ACH payments, sponsor a payment facilitator into the card networks, or hold the pooled accounts behind a banking-as-a-service product. The bank remains legally responsible to regulators and networks, so it oversees its partners' compliance, risk controls and complaints handling.
The phrase "bank sponsor" is sometimes used for the relationship itself, as in "finding a bank sponsor", or for the activity of sponsoring, but it refers to the same role. Related ideas include BIN sponsorship and sponsorship in licensing.