I don't see it as Aang showing up to offer assistance to only one child, but as a manifestation of Tenzin's inner demons. That Tenzin's were apparently far friendlier indicates how he is more progressed spiritually than the others. It's a shame, what Bumi and and Kya saw sounded fascinating, and tragic. I wish we could have explored those two more. Apparently, Bumi saw some messed up shit in his time with the military. It's touching how he can still be so happy-go-lucky.
On a serious note, Aang probably didn't actually play favorites. All the time he spent with Tenzin was more likely him studying and learning Air Nomad culture. The "vacations" were probably just pit stops on the way to Air Temples as both places he mention are near them.
In other words, Aang spent more time with Tenzin because Aang had more he could and needed to teach him. With Bumi and Kya, it's like teaching Katara Air Nomad culture. Interesting but not something they really need to know.
Plus if Aang played favorites like that, I doubt Katara would let that fly.
They have discussed on Tv tropes another reason why Aang acted like he did. tvtropes.org/pmwiki/pmwiki.php… He had no parents. He was raised in a monastry , and probaly never expected himself to become a father. "And, from what we've seen of it, parenthood wasn't something the Air Nomads were very familiar with themselves. The closest thing he had to a parent was Gyatso, and Gyatso wasn't seen to juggle attention with anyone else. So Aang was never taught parenting, never experienced it, and the nearest thing he had to a parent didn't show him how to be one to multiple people."
To be fair, by the time Iroh retired and had time to actually spend with his nephew and niece, Azula was a full-blown psychopath. Iroh was scared of her. He all but admitted it. Scared not so much that she could hurt him or Zuko, but more of the fact that she could do it with a smile.
A lot of people have been saying that it wasn't actually Aang, and rather a figment of Tenzin's imagination. That might make more sense, but this comic isn't funny if you interpret it that way so...yeah.