I'm trying to change my fill valve because for some reason it does not send water to the refill tube and only fills downwards (C-Max anti siphon), but when I go to shutoff my supply valve, turning it clockwise until tight does not seem to stop water flow. I tried counter clockwise all the way the other direction and that too did not prevent water flow. Does this mean that the supply valve has lost its grip or is there another possible explanation?
Does this mean that the supply valve has lost its grip...?"
YES. Your valve has failed and it is very common for "bib" type stop-valves to fail this way. It will need to be replaced which requires shutting off upstream (maybe even all the way upstream, as in the whole house). For a couple bucks more you can get a ball-valve type, which should be more reliable for you. I can sometimes get failed bib-style stop valves to shut off by really cranking them tightly (which destroys them forever... but it was already jacked up, right?)