Is there a term for the imaginary line extrapolated from the tip of your finger when pointing at something?
In a word: no. In comics you will occasionally see a dotted line indicating the item you are referencing. English does not have a term for that.
You can express the idea by saying, "the line extending from his/her finger toward the thing he/she is pointing at" or perhaps "line of demarcation" or such, but you have to go at it that way.
Is there a term for an imaginary line extrapolated from the tip of the finger? I doubt if there's a single term referring specifically to this line (there's no term for this like there is "equator" for the imaginary line around the earth), and so I would advise you to just use, "imaginary line".
Point at something, and if you were to shoot along the imaginary line that forms from your finger to the target, you would hit the target.
The 'axis' of his/her finger seems like a good bet. I draw from medicine for this usage, where we talk about the 'axis of the birth canal' or the 'axis of the orbit'. It's a straight, imaginary line running through the centre of a structure. So the axis of the finger would run through the finger and continue beyond its tip in the direction that it was pointing in.