No, it's not elliptical. It doesn't have the structure of a canonical clause; it consists of a wh-word (either what or how) followed by a preposition phrase headed by about, here meaning "relating to; concerning; on the subject of" (definition from Collins).
Quirk et al. (1985)† describe this as a type of irregular wh-question, beginning with either how about or what about. After these words are either a noun phrase or an -ing clause:
How about some lemon pie?
What about taking the rest of the day off?
This form of question can be used in two main ways:
With directive force:
What about following us in your car?
This could mean "Please consider following us in your car." The listener could respond "Sure, I can do that," consenting to the proposition.
How about some apple pie?
This could mean "Please consider having some apple pie." The listener could respond with "I'd love some!"
To set a topic for inquiry:
I'm fine. How about you? Are you feeling well?
This changes the topic from the speaker (I) to the listener (you). As in this example, the speaker can ask explicitly about the new topic afterwards.
Or, as in your example, the listener can answer about the new topic:
Tom: We have math, science and art. What about you?
Amy: We have English and P.E. What do you have tomorrow?
In either case, nothing is omitted.
This construction is used most commonly in conversation in informal English.
†The reference cited here is A Comprehensive Grammar of the English Language (Quirk et al, 1985). The relevant section starts on page 839.