"Who is that for?"
Fifty years ago, someone would have pointed out that:
Prepositions should never be used as the last words in sentences.
'Who', governed by 'for' although not obviously so in this sentence, should be in the accusative case and thus be replaced by its variant 'whom'.
They would probably have demanded: "For whom is that?"
Today, about 100% of native speakers would use "Who is that for?" (we tend not to ellipt when speaking to very young children), or, as you say, "Who's that for?"
If by “showing” a bottle you mean holding and displaying it, “Who's this for?” would be said far more frequently than “Who's that for?”. You may find Is there a clear delineation between the usages of 'this' and 'that' in American English? and linked questions of interest. Briefly, use this for items proximal, and that for distal.
On one hand, the statement "For whom is this bottle intended?" is a more elegant way to ask the question.
If I were to ask my own child, I would probably use "Who's this (bottle) for?", almost exactly as you've phrased it in the question. I think toddlers deserve some syntactical leeway.
You could also ask
"Whose is this?" (with or without "Is this yours?)"
The connotations are a bit different, but can be equally educational and playful for the child.