looks kind of nightshade-y.

I agree with Persimmons that it is a nightshade (Solanum sp., perhaps S. dulcamara or S. nigrum) They have tiny flowers like tomatoes but are white or purple. The berries get eaten, so it was probably planted by a bird who had eaten the berries. If you are wondering, yes, it is a weed, and if not pulled out will reseed itself in your garden.

Thank you! Another possibility - morning glory?

It definitely isn't morning glory. Morning glory leaves are much more arrow-like, and it climbs from the very beginning.
It's something I know as 'wild nightshade', so I agree with everybody else.

"Deadly nightshade" is what I know it as. Pull.