this post is a sub-topic of that post, which seems to be discussing without a common accepted foundation.
although this post has not yet got much discussion, I prefer to use this () notation to denote the sequence.
given the sequence of the prime numbers $(2, 3, 5, 7, 11, 13, 17, ...)$
and
the sequence of the even numbers $(..., -6, -4, -2, 0, 2, 4, 6, ...)$
is the intersection of these 2 sequences a number 2 or a sequence of a single number (2) or a set (I'll pass the last one, anyway, I have no the right answer, so I list all of the possibilities)?
Strictly speaking, in the context of your question, those sequences are just sets, since the order doesn't matter. The intersection of two sets is always a set, even if it happens to be a set with just one element. So you should say "$2$ is the only number in the intersection".
In practice you won't cause confusion if you say "the intersection is $2$". Better just to say "$2$ is the only even prime" with less formality.
If you really did want to talk about the intersection of sequences, what might you mean by the intersection of $(1,2,3)$ and $(4,5,1)$? Each contains $1$, but at a different place.