Remember we’re dealing with mathematical idealisations, that is, the needle has no width, is exactly the distance between the lines, etc.
The probability of the needle being exactly perpendicular to the line is zero (I won’t get technical, but the chance of it being at any specific angle must be the same, and if this was greater than zero then the total probability must be greater than one (indeed, infinite) which is nonsense.)
This means that the probability that the needle crosses the line is unaffected by this case, whatever result you ascribe to it. I personally would say it touches a line, given that it touches one, and the options are ‘touches’ and ‘doesn’t touch’.
I can go into the maths of it in more detail if you’d like.
To give an analagous example, but one which your intuition may work for, imagine we choose a number x (uniformly) randomly between 0 and 1 (inclusive). If 0<x<=0.5 say y=0. If 0.5<x<=1 say y=1. If x=0 then y 0, or (alternatively 1, or 2). The probability of y being 0 is then a half, regardless of what answer we picked for when x was 0. This isn’t rigorous by any means, but I hope you believe me