What is this type of sink drain/plug called?

We recently moved into a condo, and the sinks have screw-in drain cover. We are trying to find a replacement drain cover that is able to close, but I can’t figure out what the type of drain is called to see if this exists!

The drain itself has a plus-shaped brace that sits about an inch in, with a threaded hole in the middle. It looks a lot like a bathtub drain, but the threading of the hole is slightly different.

The cover is a wide metal disc with a short bar coming from the bottom which is partially threaded. I feel like this is the type of drain + cover in a lot of hotels.

EDIT: 7/8

I thought it might be a ‘pop up click’ or ‘click clack’ drain (as two answers suggested), and ordered a pop up stopper and bolt, but the length of the bolt is way longer than our drains (even in ‘compressed’ position).

The length of the stem on our drain cover is ~33mm, with ~13mm thread, while the ‘pop up’ we got has a 66mm bolt, with much larger thread. See image below for comparison.