Let's get this out of the way: We're talking about an April Fools article that focuses on cats and that, despite its publication date, predates the 3.5 revision (for example, look at the cheshire cat's special ability feats). A DM that allows a PC that's a catfolk (Races of the Wild 92-5), hengeyokai (cat) (OA 10-11), tibbit (Dragon Compendium Volume 1 21-5), or whatever should expect weirdness if the article's rules are used without modification!
The Handle Humanoid skill doesn't allow the feline to try again to teach that trick to an appropriate creature if the feline fails the Handle Humanoid skill check or—if the check's successful—if the feline pauses in its efforts to train the appropriate creature. In such a case, a different feline must attempt to teach that appropriate creature that trick.
Similarly, the skill Handle Humanoid puts no Intelligence-governed limit on the number of tricks an appropriate creature can be taught. All appropriate creature have a limit of 6 tricks that can be taught to it using the skill Handle Humanoid.
Given these key differences, this DM would keep separate the tricks taught to a giant, humanoid, or monstrous humanoid that has an Intelligence score of 1 or 2 by a handler with the skill Handle Animal and a feline with the skill Handle Humanoid.
While on an adventure, Hicc the cloistered cleric 17 of no particular deity rolls a 1 on his Will save against the 5th-level Sor/Wiz spell feeblemind [ench] (PH 229-30), which, among other effects, reduces his Intelligence to 1. After escaping the dungeon, the other PCs realize that it's usually Hicc who would fix such a problem and usually Hicc who would get them home. They'll be spending months in the wilderness before reaching civilization.
Ekaj the totemist has ranks in the skill Handle Animal, and the Handle Animal skill can be employed on creatures that don't possess the type animal and that have an Intelligence score of 1 or 2, but such skill uses see their DCs increase by +5. Ekaj sets about training Hicc for the purpose fighting.
Vitras the tibbit dread necromancer has ranks in the skill Handle Humanoid. He's already (and perhaps even without their knowledge) trained most of the party in the tricks defend and open!, but now he sees his chance to train Hicc (who'd always been smart enough before not to spend that much time around the tibbit) some tricks… like that trick fetch!
The two skills, while sharing a superficial resemblance, in this DM's opinion, don't overlap. Tricks taught using one skill can't be activated by the other skill, and tricks taught using one skill don't count toward a creature's trick maximum by the other skill.
By the way, the skill Handle Humanoid breaks campaigns
As written, the skill Handle Humanoid essentially allows a feline to take a full-round action to dictate the behavior of a giant, humanoid, or monstrous humanoid, and that creature can't resist. Whether this is for only the subject creature's next turn or several turns, this casually removes such creatures from the action. Although there may be some real-world truth to it, in a long-term campaign, the skill makes felines the world's masters. (Whether felines are the world's secret masters will depend on the campaign!) For example, felines will be capable of acquiring free material support (via the feed me trick) and items of all kinds (via the fetch trick!) from the creatures that most commonly build and (seem to) rule towns. A campaign that allows the use of the skill Handle Humanoid unmodified should have strict feline controls in place or see felines treated as gods lest the more experienced felines casually order their giant, humanoid, and monstrous humanoid minions to their doom (or, more likely, to their more amusing utter humiliation).