Silver was smart, successful, and beautiful...
She was dating Bruce Wayne. Batman showed up at a charity gala she was throwing, and she saw him in person. So she recognized him. How? The way any person of average intelligence would. Someone you're romantically involved with comes into a room wearing a mask, talks a bit, and you recognize them. If you can't recognize the person you're dating on Halloween, that's a relationship that was never meant to be.
Silver's words were "Oh my god! It's Bruce!"
That's how it would be. No nonsensical puzzlement, no need for sleuthing. She was in love with him, she saw him dressed as Batman, and the Batman costume didn't fool her. It was the most obvious, natural story event possible, the one that comic writers had worked like demons not to use. So Englehart did.
Then she saw Batman actually in hand-to-hand for the first time, fighting Joker in the rain, and again she had the normal person-of-adequate-intelligence reaction: It scared her. Every bit of it did. She didn't want to be in a relationship, where who she was dating at the time, would do things like that. So she left Bruce, constantly due to how she feared he was risking his life as Batman.