Glad you asked. ;) I hit 28 last night and summoned my first Tiger Familiar, and let me say wow. ;p
I would say that they can contribute quite well in battle only if they are still similar lvl to you. We were fighting the Yhoator Mandragoras in the 2nd Kazham jungle which as everyone knows take bloody ages to fight and just drain everyone of mp and eventually patience. ;) Anyway, the jug for Tiger Familiar is lvl 28 too, so he's basically my lvl, and oddly enough fights pretty well against these. He was doing good dmg, like a warrior of comparable level and even has double-attack sometimes or so it would seem. I would speculate that using the lvl 23 HareFamiliar would have been worse off since he is much lower lvl. His dmg output would be smaller. And by 28 you get "Sic" so I was able to let my Tiger unleash some nice specials.
Be warned.. we had a White Lizard near us which don't aggro but are really **** hard, and I told my Tiger to Sic, and he did an area of effect special, aggroing the lizard. Miraculously, he killed the tiger and left us alone. (he was doing 100 dmg hits on the tiger). :)
Anyway, as I lvl up, I guess the Tiger will be less useful as I get higher because he won't be high enough to do well against the stuff I'll be fighting. But we get new familiars anyway as we get higher. Or.... are they summoned at a comparable lvl to me. Hmm, I'm not sure on that. Maybe the 28 thing is just a restiction on how soon I can use the tiger, and doens't necessarily mean he's lvl 28. Wonder if anyone knows.
Carrot broth is for Hare Familiar, it's like having a little Rarab fighting for you. ;) And yes while that's not so exciting, having a Tiger that's as tall as me and watching it use its specials is very cool indeed. Gives a whole new meaning to the "party trick". :P It also doesn't kill their xp at all since it's your lvl or lower. You just get the 30% cut as always. But the Tiger sped up the fights against those onions big time, allowing for chains which we weren't able to do before. :) It's like having a 7th fighter in the party.
Hope that helps. :) I bought 12 jugs of meat broth (tiger) for 2,500 gil. But they usually go for 3,000. I was lucky.