Infinitely many, though in a plane, only one straightline.
However, an arbitrarily large number of curved linescould pass through two points.
[As an aside there some non-Euclidian geometries in which the above do not apply, but I'm sure they lie outside your scope of inquiry.]
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There is only one possible line that can can through two different points, presuming there are no overlaps.
If you're talking about straight lines (not curves) the answer is one.
Since the question does not require them to be straight lines, the answer is infinitely many.
An infinite number of lines can be drawn through a single point, but only one through two points (of course, if the points don't have the same coordinates).
The two lines are identical.