The socket appears to have overheated because the metal seems to be blackened, and also I see some corrosion on the rivets. That could be enough to prevent the normal flow of electricity. WITH THE POWER OFF, you could try cleaning the socket contacts with something like a dremel fitted with a small wire brush on medium speed. Also, over time the bottom contact (the little tang in the middle held with a phillips screw) can get bent out of shape. Still with the power off, use a small screwdriver and pry it up slightly and gently.

If all else fails, on a regular fixture the socket is fairly easy to replace, but I have no experience with pool lighting. Judging by the picture it seems that it is sealed in some kind of resin or rubber. If that's the case, it might be easier to replace the whole fixture.

What kind of tester are you using? You want to measure voltage between both contacts. The bottom contact is the hot, and the threaded part is your neutral. A non contact tester will only tell you if there is 120V on the bottom contact, but won't tell you if the neutral is open. A decent multimeter can be had for around $20 on Amazon.