Advice on how to remove a faucet cartridge
It looks like what we call a "Delex" cartridge; the original version was manufactured by Delta but there are knock-offs using the design now. 

Once the retainer nut has been removed, that cartridge is just pressed in; keep yanking directly upward and it will come loose. I sometimes craft a "pull handle" with a flat piece of steel (like a crossbar for a light fixture) and screw that to the stem using the handle screw, giving me something to pull on with both hands.

Don't make the mistake of just swapping out the cartridge; the actual seal washer is below, with a spring under it.