A couple of my bikes have white tape, one has yellow, they both show dirt, which tells me that my black tape has to be dirty too, so I clean them all.
I use Dawn for Dishes, not the citrus crap though, with a sponge, and lather it up well and scrub the tapes, and they come clean; on rare occasion I might have to use 409 to get stubborn stains off.
I will say this, I can ride for a lot longer than a couple of dozen rides and my tape won't look near as bad as yours, not sure why, though it could be that I also wash my gloves about every 4th ride, and by washing my gloves it could be keeping the tape cleaner longer.
I haven't found any tape, including Fizik tape, that will stay clean. I think the old 70's and 80's era plastic tape stayed cleaner than the new stuff, but that plastic tape had no cushioning, and it didn't breathe so your hands sweated all over it which made it slippery when it got wet.