Have you checked to see if any of them are bad pixels?There are single dots in a few of the photos that I have checked. I'm sure they are just being repeated in the stacks, but they end up being lines because of the different focal planes.
It sounds as if they may not be bad pixels and this is indeed a stacking artefact. I suspect the problem is caused by the sharpness of the tiny point of light, which makes the software think they are genuine tiny sharp areas, which therefore need to be included in lots of focal planes.
It would be interesting to see the stacking result for this photo from the best stacking software. I know you tried Photoshop which is OK but you should really be trying Helicon Focus. I think you could try the free trial. It has several stacking methods you can try.