Oh yes, advertising often takes liberties with grammar and usage for maximizing audience recall of the intended message. In the particular case of the video commercial you presented, however, I don't think that using the pronoun "me" instead of "him" or "her" is an inadvertent grammar mistake in the song lyric "can follow what you tell me." Keep in mind that this particular line of the jingle being sung by the kids is actually an extension of this statement of the woman in the beginning of the video: "Is your child school-ready?" It's logical to conclude that the statements that follow this question are actually organic answers to that woman's question using her point of view (POV), except that the kids are singing the answers interpretatively (in the sense that the kids are taking that POV as their own, reciting the lyrics verbatim). Thus, it's really not a grammar error to use "me" (possessive referring to the woman) rather than "her" (possessive referring to nobody in particular in that situation). Another way of looking at this is, of course, that the answers are being made by parents since they are the ones directly addressed by the question and that the kids are speaking on their behalf, but I'm afraid that this will just lead us to an even more abstruse grammar discussion that will only lead us farther from the basic point that needs to be emphasized here.
The deliberate use of mildly debatable grammar indeed may sound problematic to people uninitiated in the creative aspects of advertising, but from my own experience in communication, it does substantially enhance audience recall of the message being presented. The fact that the seeming error caught your attention and you took the trouble of calling my attention to it is, in itself, an indication that the message got through to you and made you want to correct and refine it somehow. This, I must say, is a big bonus to the ad's communication effectiveness, one that doesn't detract in any appreciable way from the quality, clarity, and persuasiveness of its message.