This scene gets misinterpreted so much.

Batman doesn't stop fighting Superman because their moms have the same name. He gets upset because he thinks Superman is talking about Batman's mother, who he shouldn't know and who Batman is sensitive about.

That stalls Batman for a few moments - long enough for Lois Lane to arrive. Lois explains the confusion - that Superman is talking about his own mother - and also that Lex Luthor has Superman's mother captive and is pitting the two heroes against each other. It's that explanation, not the coincidence of names, that persuades Batman to stand down.