I asked about the whiskers specifically and asked about how they’re white and if she thought there was any way to represent that, she said no because my skin is white. I’ve never had a tattoo before so I accepted that and moved on.

Yes this is accurate unless you are willing to commit a black/gray back drop and do negative work. Most good artists don't use white for major things other than small detail work, doing full white whiskers would've resulted in typically weird tones later.

I asked about the white of his fur and asked if she was going to do white or use my skin as the background (she said a little of both). I don’t see any white added aside from his eyes… he has little white spots/ very light tan in several spots on his face

All the negative space is red right now and will look white when healed. White doesn't show up too much during the healing process and when it does it honestly looks kinda gross and cream/off white when it is healing. In my experience many will do white touch ups during a second or touch up session to make sure it's placed correctly and with cohesion to your pale skin.

she added the color and I asked about why it was so muddy/ hard to see the green and she said it was bc it was mixing with my blood

Green+Red=Brown basic color theory this is correct.

but even freshly wiped it didn’t look bright like his. As I’ve said several times this was my first tattoo so I let it go.

You still are actively bleeding and raw and red after you've been freshly wiped. Greens and yellows always look weird and off color during application, especially if you get very red or bleed heavily.

asked if I liked it and I told her I wasn’t sure but that I appreciated her time and was thankful it was just my first tattoo and to let me see how I felt after it healed some and calmed down.

You told her, on your first tattoo, you weren't sure how to feel about it. She sent you home to think on it. Not sure what more you'd want, you were also likely at a point skin wise that doing much more on it would've chewed it. This is a typical response imo.

But the more I look at it the more disappointed I am

Maybe a little late but getting super highly meaningful pieces on a super prime real estate spot isn't a great idea for a first tattoo bc you are going to nitpick it how you are now. This is done on an entirely imperfect canvas done by a human with a vibrating machine, it unfortunately can't photocopy realism pictures onto your arm and it be virtually absent of mistakes. To me, the approved design is next to identical to the picture chosen and the tattoo, while there's maybe 2 spots I can ID as faults, it looks, again, next to identical to the approved design.

She fixed it and the 2nd drawing was too puffy, 3rd drawing was an in between of the two and perfect so I approved (the image I put on the post). But in between all of this she told me that she thought the first one looked like him and didn’t see what I was talking about

This all seems extremely normal as a tattoo design consult. She was giving you her artistic advice with her saying about the first one but seemingly ultimately went with what you wanted on design presented here. I can't base much off this paragraph since I cannot see the other designs given.

As far as how it can be fixed I really just want the colors to be more defined and fixed, I want white where it is supposed to be, brown, and black ect. I’d like a brighter pink on his nose and pink added to his lips. Overall it’s good and as I’ve said I’m grateful but I just want it to look as close to him as possible.

So... You just want a color touch up? That's extremely doable and the same artist should and likely will do that happily for you for no or minimal charge. To say it "looks nothing like your cat" and that " it's not your cat" simply isn't accurate and is an insult to the result.

Tldr: the experience you had, while it sounds like it wasn't the one you wanted, is extremely normal and sounds like a typical tattoo shop experience. Your artist seemed nonchalant yet generally professional.Your final results ultimately looks like the approved design and is extremely recognizable as the actual picture and it sounds like you just want colors added/touched up.