The detergent or soap is not added to make the neem spray work. It's added to make it possible to mix the neem oil with water when preparing neem spray.
You probably know from experience that oil and water do not mix. Oil will float on top. If you do not add a detergent then the neem oil floats on the surface and you end up spraying only water. (The neem oil will in the end probably stick to the walls of the spray bottle.)
So, you need some detergent every time you want to dissolve neem oil in water.
When using neem as a personal insect repellent spray you obviously don't need insecticidal soap. That soap was specifically developed for plants.
Any soap acts as a detergent. For a personal neem spray use a soap developed for people. You can use any castille or liquid soap. Use something that you don't mind putting on your skin.
Make sure that all the neem oil dissolves and none is left floating on the surface. If necessary, add more soap. (Using warm water helps.) And don't forget that neem oil is not stable when mixed with water. Only mix a little at a time, ideally only as much as you intend to use.