Confusing but I think you are right, they are the same thing. BNY Mellon owns Pershing. (Don't know about current times, but Pershing used to perform brokerage reporting services for some big companies like Vanguard. Similarly BNY/Mellon does banking services for many brokers. So you checks from ABC Brokers may actually say (or be cleared though) BNY/Mellon bank.
https://www.pershing.com/about/our-businesses#:~:text=Pershing%20LLC%2C%20member%20FINRA%2C%20NYSE,a....
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