it's a single hole faucet.

single hole bath faucet

Bronze single hole bathroom vanity faucet:

https://www.google.com/search?q=Bronze+single+hole+bathroom+vanity+faucet&source=lnms&tbm=isch&sa=X&ved=2ahUKEwjF38PbhvPmAhXMm-AKHY4OAGEQ_AUoAnoECA0QBA&biw=1366&bih=628#spf=1578453836779

Would a single lever work rather than having the two knobs? Finding one of those might be easier.

Yes to push pop-up drain. I tried one, liked it so much that I switched out the drains in the other bathrooms.

This is is an image of a brass faucet. Don't know if they have it in the specific finish you want ? bronze ? but there is a place to click to see a long list of 27 finishes on the right.

https://www.torrcodesigncenter.com/Hartford-Stamford-Danbury-Fairfield-New-Haven-Waterbury-East-Windsor/Newport-Brass-932-10B-Oil-Rubbed-Bronze-Single-Hole-Bathroom-Sink-Faucet.HTM

If you want two handles, search for 'single hole faucet with two handles'.

The lever is easier but there is no place for it to operate-- the faucet is jammed up against a 5 inch tall built-in backsplash type thing. Thanks everyone! I had looked but clearly not well enough!

Vintage Tub and Bath has a bunch of them. You can get a lever on top that goes up and down above the faucet. They also have the ones with two handles.

We have a push pop-up drain and love it.

I haven't closed the bath sink drain in forty years. . Get a pop up if you must : )

I love your faucet!! Where did you get it?

I love my Kraus pop up drains.

@Me AGirl -- the faucet came with the house. It leaks in 2 places. I suspect it's from 2000, ish?

And regarding the pop up drains-- do you just stick your hand through the dirty water to pop the drain up? I can't say that's appealing. I do soak laundry in the sink at times... No many other use cases for closing the drain.