Unfortunately, the short answer to your question is that you can't change the colour of paths in Photoshop.
The path colours in Illustrator are used to distinguish objects that are on different layers, every layer gets its own colour. Paths in Photoshop are not attached to a particular layer which I guess is why Adobe don't feel the need to provide similar functionality. Plenty of people have requested this feature going back years, but for whatever reason it is still not available.
The visibility issue is (mostly) handled in Photoshop by the fact that the path colour is dynamic. It appears darker over light regions of the image and lighter over dark regions. The performance of this feature has improved in recent (CC) versions, but it's still not perfect. I remember in CS6 and earlier it was particularly difficult to see paths over grey mid-tones.
There is a workaround if you are struggling to see paths over a given image: You can add a flat colour fill or adjustment layer over the top of the image to lighten (or darken) the image enough to make the path easily visible and then delete this layer once you have finished drawing your path.
Not ideal, but it will get the job done.