I have a few items looking like the following but am unsure as to what it is and what is used for. Any help appreciated.

It does fit in my press where the shell holder would go.

Male part of a cast bullet sizing die.

And just what would the female side look like?

Would this be a setup where you size, not lube, a cast bullet one at a time using your single stage press??

Lee makes these, but I don't recall my Lees having caliber stamped on the bottom. Maybe the newer ones do?

The "female" looks like a die that screws into your press (yes, single stage) with a hole through it and a size designation stamped on it above the threads, along with a locking ring. You adjust the die so that when a cast bullet is placed on top of the "pedestal" installed in the press arm, it gets pushed through the sizing die.

I like the Lee and NOE push-through sizers, particularly with bulk powder-coated bullets. Easy to size a few thousand in an evening. Also works for seating gas checks.

https://leeprecision.com/bullet-casting/bullet-sizing-kit/

Their male part looks a bit different than the pic above.

Could be shop-made bullet pusher. I make mine similarly, with the heads turned to fit neatly into a .45-70 shell holder.

Thanks to all for the helpful information!

It looks like Lee uses one generic picture for all the sizing dies on their website. Here's one that's similar to the OP's pic.

If you go on amazon and look at the .314 one it looks exactly like the one the OP has and the one shown in Jim's post above.