The question is concerned with what many call nonconstituent coordination and hence abbreviate as NCC. Certainly there are other terms denoting the phenomena as pointed out in the comments. In the literature, such instances of coordination were, I believe, first discussed in an article in Language by Richard Hudson in 1976. Anyone who has ever worked on coordination has come across them and had to offer some sort of account of how it is possible for such nonconstituent strings to be coordinated.
The claim that such instances of coordination involve ellipsis is refuted by examples like the next ones:
(1a) We gave nobody [chocolates today] and [flowers yesterday].
(1b) We [gave nobody chocolates today] and [gave nobody flowers yesterday].
If ellipsis were involved in (1a) in terms of an elided gave nobody, then (1a) should have the same meaning as (1b) in which nothing has been elided. But that is not the case, for (1a) clearly has a different reading from (1b). The negation in nobody in (1a) scopes over the entire coordinate structure, whereas in (1b), the negation in each nobody only scopes over the conjunct that immediately contains it.
Another indication that ellipsis is not involved in such cases is that the intonation contour is normal. There is no special contour of the sort associated with right node raising (RNR, e.g. You like, but I dislike, writing syntax papers) which can more convincingly be viewed as involving some form of ellipsis.
One important conclusion I think should be drawn from the existence of such instances of nonconstituent coordination is that coordination is not a good test for identifying constituents, despite the fact that it is widely used as such. It suggests that far too many strings are constituents, strings that other tests do not verify as constituents. It is very much unlike most other tests for constituents in this regard.
To answer the question as directly as possible, the reason why such strings can be coordinated is that coordination operates on parallel strings, whereby these strings need not be constituents. As long as the strings are appropriately parallel, they can be coordinated. In a dependency grammar framework, strings are appropriately parallel if they contain matching roots, i.e. words that are not dominated by any other words in their conjunct. I can elaborate further and provide references if anyone is interested. Contact me at firstname.lastname@example.org.