2Emily: That too, though 1st things 1st I was curious why people can't say "me & my friends" if they can say "me too" (instead of "I too"), "that's her" (instead of "that's she"). Many posters I observed writing this, f.ex. Rocker said before "me & my friends". What filledeplage said is right but wouldn't it be correct when smb. says "I grew up to be shortie" to reply with "I too", as it agrees with the previous human? Why it's suddenly "me too"? If they can use it, then by logic I assumed that "me & my friends" is allowed as well. or maybe it's dumb logic. Carry on.
Both of those are examples of convention defying what's grammatically correct. If you are saying, "that also applies to me," then "me too" would be correct. But if you are saying "I think so too," then "I too" would be correct. Of course the convention is to use "me too" in either case; not sure why.
In the case of "that is she" vs. "that is her," "that is she" would always be correct, but is never used. There are examples with "this" where the structure is used correctly, though. Most people on the phone, if someone says, "may I speak with Emily?" will answer "this is she" rather than "this is her." In that case, they would be being both conventional and correct.
My dad always made fun of his high school English teacher for a really, really bad 'poetic' saying she made them memorize, though I'll note that it's effective enough that he remembered it, I remember it, and my daughter probably will, too:
We may all laugh at he
Who takes an object with the verb to be. (This is the first time in my life I've wanted to use a vomiting emoji).
Consequently, whenever I called him, I would say "'tis I, papa" in a bogus English accent. But, yes, the correct but rarely used form is "it is I" not "it's me."
My guess, for the "that's her" construction, is that it seems correct. I think people confuse it with a standard subject-verb-object sentence: "I tripped her," "He kissed her," etc. So if you don't think through the verb To Be being used as a linking verb, "that is her" sounds right.
Just my guess.