As you can see here the fence gate will connect to a cobblestone wall however it will not connect to the normal fence.

How To Reproduce:

1. load a 1.12.2/1.13 pre 6 snapshot world.
2. place down a fence gate next to a cobblestone wall (it should connect)
3. place down a fence next to a cobblestone wall (it will not connect)

Feel free to mark this as a duplicate because it likely is one. I searched for the same problem and couldn't find anyone else who reported it.