Having a similar issue. I haven't bred horses yet but I did breed a mule and had a similar issue. Here's a video of the glitch: https://youtu.be/Bc0bwaGw0zA
It seems mobs and players just can push baby horses or mules through the fence at the corners and at gates. This is on Xbox One / happened with Windows 10 version too.
Have tested with Windows 10 and Xbox One with multiple players; does not work. A lot of achievements are taking a while to register or taking multiple times to register as well, but this one never came.
For Xbox One and Windows 10 this is happening with the new update still. Sometimes after breaking swamp trees, the vines will prevent some leaves from de-spawning. Maybe it would be possible for leaves and (or at least) vines to break down like Chorus Plants when all wood of a tree (or when the top block holding the vines if not connected to a valid block) is destroyed.
Screenshot of issue (Xbox One): https://1drv.ms/i/s!Ald2cMebzOfcjlVEdXWX58lvr2S1