Stars are so far away that their apparent width is essentially zero when compared to any pixel of a camera or TV screen.
And yet we can still see them.
In an electronic sensor (CMOS or CCD for example), the pixel sums all the light that reaches it. Even if light only falls on a small part of the actual sensitive area of the pixel, it will generate photocurrent that the camera will record as part of the image.
Of course if the image of the star on the sensor is small enough, it might fall in a gap between the pixels. But there are also resolution limits to the optics forming the image that generally prevent this being an issue.
According to our eyes stars have a finite albeit very small width. Why is this, and what is this width?
This is the resolution limit of our eyes, determined mainly by diffraction from the aperture of the pupil of the eye.