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If the cat dies, and you didn't reload, you don't get the opportunity to keep the cat as a familiar. but I can 100% confirm that I now have summon cat familiar on my bar. It runs away during most combat, but focus on escaping without it dying before you go back into town and complete any other quests to minimize the chances you don't lose it. Heal it when it's low health from poison, etc. and as soon as you escape with Pet Pal, it will start a conversation about it's previous owner. You have to be a good owner while it follows you in order to keep it. :)
I have a slight problem...
I killed the cat in order to speak to buddy because it would not stop following me, now the cat is dead and I think it might have been a bad idea, is it very important later on?
Well i lost the cat somewhere between hour 30 and 35. The problem was I exited fort joy via the sewer after killing Kniles without the cat following. You can teleport to waypoints to temporarily lose it and stop it following you, useful when talking to Buddy the dog or taking on a fight with lots of area of effect damage etc.

I THINK, If you exit fort joy and get the quest update without the cat following the cat seems to stay in it place. I stumbled on it again after hour 48 at the scaffold tower exit near Orviand. x310 y124 (I was chasing after Gawin trying to kill him).

Maybe check all the fort joy exits before you consider the cat dead, since it is a very early summon with a utility teleport ability. Sewers, Scaffolding, Docks, Bridge.
I just now met Orivand with the familiar, having escaped previously with the cat and tested a bit.

The only thing I haven't tried is summoning the familiar before the fight and re-summoning it every time it expires during the fight until it is over and the familiar is still there. But apart from that, I had no special dialogue, and in the end, the Swap skill was unchanged, it still only worked with the summoner.

In short, for what I can tell, there is no upgrade to it. Not just by going to Orivand after escaping with the cat, at least.
Went into the alcove and it died to the poison, dropped me claw, intestines and earth essence 1 each.
Here I was just minding my own business while Ifan had a little 'chat' with Magister Borris and one of the archers just starts shooting my cat! What the hell!? Had to reset to before the croc fight to get it back.
That dude just starts sniping the cat if he see's it! I saw red and went into a murder frenzy.