I don't think that's correct. It's true that if one did, in fact, continue the truncated sentence it would be "I," as in "She has many books like I have." Or perhaps, "You should do as I have done." In that case "I" serves as the subject of the second phrase of the sentence, just as "I" serves as the subject of the first phrase of the sentence. But if you don't continue the sentence by adding the second phrase, then "like" isn't used as a conjunction. So the reference (me/I) is the object of the phrase, not the subject. "She has many books like me." That's correct. "He's handsome like me" (correct) vs. "He's handsome like I" (incorrect).
No, your grammar is incorrect. If it's "many books" or "as many books" or "is good looking", then the person you're comparing to is "I", not "me". I have as many books as she (does). She has as many books as I (do). I am as pretty as she (is). She is as pretty as I (am). NOT Me has many books nor Me is handsome.
So traditionally, yes, it should be I, not me. But English language speakers across the globe very comfortably say "as many books as me" without any discomfort or impression of incorrectness.
In the general shift of everything across the incorrect-informal-general-formal-archaic spectrum, using "as I" has left behind general and has become formal, presumably headed to archaic soon.
On the other hand, "as me" has left behind incorrect and become informal, presumably to become general soon.
You are absolutely right, but the reason I didn't mention it up is because "alto" is an adjective, and I was focusing on this question, which is describing when tanto is used on a plural noun, "libros." I wasn't trying to confuse anyone by listing every single possible situation. Nevertheless, the point I was trying to make was still that there is a direct comparison present when using "tanto" and "como," not just a simple similarity but exactly the same in quantity or quality (the EXACT same number or books or the EXACT same height measurement)
If you want to be general, the rule appears to be:
Tanto (plural noun) como = "as many as" Ex. as many books as... Tanto (adjective) como = "as (adjective) as" Ex. as tall as...
It will vary, depending on where the speaker learned Spanish. Pick one and stick with it. If you are in the Americas, unless you are in Argentina, using the 'y' sound like in English is probably what will sound the most natural to the locals.
The other sound (which is usually not actually like a 'j' sound but something more like combining the English 'z' with 'sh'-it just gets described that way because the sound doesn't exist in English) may sound a little silly to some. Either sounds should be understood by all Spanish speakers.
The recommended translation within the exercise was, "She has many books like I." I don't believe this is grammatically correct in English. I believe it should be "She has many books like me." Interestingly, on this discussion page, the Translation is, "She has as many books as me." Does the addition of "as" change the reference from a subject (where "I" would be proper) to an object (where "me" would be proper)?