I'm replacing the under-sink pipes, and I'm confused what connector this is
That's a flared fitting (see: flared fitting copper).  They are meant to accept the end of a piece of copper pipe, flared using a special tool.  Looking at the picture, it does seem that you might be able to replace that part under the sink with a brass fitting (see: brass male-male pipe nipple ) that will properly accept your hose, but it might be difficult, given the corrosion.