Honestly, because you are asking at such a high level--just "dependency grammar" or "constituency grammar", rather than a specific grammatical theory, this question is likely not really answerable. That being said, I think you have some assumptions in your question which have lead to your confusion.
It is usually assumed that this specific type of structure is a kind of ellipsis, where a portion of the sentence has been omitted. Different syntactic theories will take different approaches to answering the questions of what is omitted, and how we end up with what we see. Just saying "dependency grammar" or "constituency grammar" is not specific enough, as there are many dependency grammars, and many constituency grammars, many of which offer possible theories on what form this particular sentence takes, and thus how to draw a tree of it.
It is helpful here to work backwards. Consider the general context where a sentence like Me too occurs. It is an addition to an answer to a question:
- Do you want cake?
- I do.
- Me too.
Answers like (2) are themselves usually assumed to be elliptical structures as well, paraphrasing the question they are responding to. For instance, (2) could be interchanged with (4):
- I do want cake.
Similarly, (3) could be interchanged with (5):
- I do want cake too.
In fact, we could say that there are a number of possible ellipses that could occur, which we might diagram like so:
- I ((((do) want) cake) too).
With each set of brackets enclosing more and more of what can be ellipsed, all the way from the full, un-ellipsed sentence I do want cake too to the fully ellipsed sentence Me.
So I think part of the issue where you're running into problems is that the predicate here do want cake (or whatever it is in context) is simply not there. There is a long body of literature from a variety of approaches on how to deal with this, but it will entirely depend on which specific dependency or constituency grammatical theory you are operating in.