Double words in a sentence, what is this called?

Solution 1:

Linguists use the term reduplication for repetitions of words or parts of words.

First, however, it's much more important in other languages than it is English, in which it's generally thought of as unremarkable or at best, a peripheral phenomenon. In Chinese, for example, 人 means person and 人人 means everybody.

Second, except under the broadest definition, which is simply the repetition of a word or part of a word, your examples wouldn't qualify because the repetitions are basically a coincidence or a byproduct of normal use of the language. Compare to

Oh! You live in an apartment? I thought your house was a house house.

In this example, the duplication is employed to contrast a prototypical or "real" house with some other kind. This has been called contrastive reduplication. Since it is a morphological device that is employed with a wide range of words, some linguists have given it at least some attention. Another example: A: I want to eat. B: Here's a cracker. B: No. I mean I want to eat eat. (That is, "Really eat; eat something substantial; eat a meal.)

See https://wikipedia.org/wiki/Reduplication

Solution 2:

I'm afraid it's not the case that every possible concept has a single accepted representative term in the English language. Double words as you described them is as appropriate as anything else we could cook up.