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:
load a 1.12.2/1.13 pre 6 snapshot world.
place down a fence gate next to a cobblestone wall (it should connect)
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.
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Duplicate of MC-27428.
Please search before creating a report to make sure your issue hasn't already been reported.

But that "cobblestone walls connect to fence gates" is a bug.See MCPE-12394. It's in the MCPE project, but I don't believe if it's a bug there, it isn't a bug here
I'm wrong. Sorry.

OK, no offence Matthew Hunter. But do you even know how to read? I clearly stated that I had searched for the problem before I reported it and you still went along and said that I needed to search for the problem before posting. The last line just above your comment even says "problem and couldn't find anyone else who reported it" I even mentioned that I knew it was likely a duplicate of another report. Geez if you've made it to this line than maybe you really can read.