Your friend understood like in your sentence as the discourse marker employed in informal English either to keep a sentence moving while you gather your thoughts or to emphasize what follows. This like is not employed in written English (except of course in dialogue).
That's really the only interpretation like will bear in this position. 'Lexical' like, which you tell is what you intended to express, is an adjective used to compare its 'head' (the nominal which it modifies) to some other entity which follows like. That second entity is what comes after like as its complement.
a HEADman like COMPthis man
So when you write
... ask like this question ...
you have an adjective phrase, with like and its complement this question—but you don't have a head, a nominal which the phrase modifies.
What you need to do is not delete like but add the head:
Many people would ask HEADa question like this question: why am I here?
And that can be expressed more simply like this:
Many people would ask a question like this one: why am I here?
Observe that this is not simply a matter of moving question. There are two questions involved here: the question which people ask and the question which the first question is compared to.