have the issue with you; recently redid my bathroom, and saw this random tap near my new shower set, exact same one as yours. If I had known, would have told contractor to place it at a different location where its not so obvious. Now have to live with this eyesore every time i shower.

Stop cock term is use to stop the incoming water to a specified section like the entire toilet.

But your is use to switch on the shower? than it should be called water pipe valve.

The two higher "tab" my contractor use the term stop cock. This type better use the conventional rubber to control the valve. The tap are using Ceramic valvel.

what is the difference between stop cock and water pipe valve?
sounds like the same thing; used to stop any water from going to the shower.

i would also like to find a less ugly and more minimal tap fitting for it if price is decent.

Is actually the same, but when you tell shop about stop cock, they usually give you not so nice looking and rubber valve. The water pipe valve got more design and can can have ceramic valve which only need 1/2 turn to switch off or on.

It is the pipe size and the tap, stop cok etc joint size. If you buy the plumping parts, you need to tell them the pipe joint size.

For rule of thumb, main inlet till tank is 3/4". While those toilet and kitchen is 1/2".

The water coming to your toilet from the water tank is call outlet. Which mostly is using 1/2" pipe. While from the meter outside to the water tank is 3/4" pipe.

The stop cock in your bathroom is to stop the water coming to the toilet tap. Since the pipe is coming from the tank and to your toilet, it is an outlet pipe. And size 1/2".

You cannot simply replace the knob. Each brand have it own design. Each brand can replace back it own brand parts. Unless those common parts. Ex. rubber washer.

Maybe those expensive type like grohe, kohler etc have an option to change the cover and knob design. But still have to using back it own brand.

So you have to hack it out and replace the whole stop cock.