When fence gates are attached to cobblestone walls, their models are moved down a little bit. However, the texture is only updated when the fence gate is facing south (take a look at the images 1-4). Let me show you the file fence_gate.json, stored inside assets/minecraft/blockstates:

        "facing=south,in_wall=true,open=false": { "model": "oak_wall_gate_closed" },
        "facing=west,in_wall=true,open=false":  { "model": "oak_wall_gate_closed", "y": 90, "uvlock": true },
        "facing=north,in_wall=true,open=false": { "model": "oak_wall_gate_closed", "y": 180, "uvlock": true },
        "facing=east,in_wall=true,open=false":  { "model": "oak_wall_gate_closed", "y": 270, "uvlock": true },
        "facing=south,in_wall=true,open=true": { "model": "oak_wall_gate_open" },
        "facing=west,in_wall=true,open=true":  { "model": "oak_wall_gate_open", "y": 90, "uvlock": true },
        "facing=north,in_wall=true,open=true": { "model": "oak_wall_gate_open", "y": 180, "uvlock": true },
        "facing=east,in_wall=true,open=true":  { "model": "oak_wall_gate_open", "y": 270, "uvlock": true }

As you can see, since south is the default rotation, it doesn't have the uvlock tag set to true. This means it does update the texture UV mapping when the block model is moved, differently from the others, where the texture remains in the same place. If you look at a normal fence gate and a wall one, you can see the lines are aligned, except on south rotation, where they are moved together (see image 5).

The solution to that would be completely removing the uvlock tag for the in_wall=true states. It would make the behavior for the non-south ones a little bit different, but it's necessary in order for the bug to be fixed.

I did a resource pack removing the tag and no problems were created by it (see image 6). If you want, you can download and check it here.