Awesome! I ran my bug reproduction script for issue #29190 against your branch and the saved animation finally looks the same as the plt.show() version.

I ran a pixel-to-pixel comparison script, and I'm impressed.
The only diffs I got were from the direct plot overlapping the gray part of the right border, while in the saved GIF it doesn't. I am attaching, below, the biggest diff pixel-wise (2 2 12), from frame 54. This overlapping could be a result of @QuLogic's changed transparency handling. However, I don't think it's something users will care about, plus the fact that the border overlaps into the graph is an issue unrelated to his changes.
I also got diffs on the axes' numbers, which I masked out. But I don't think they are noticeable either.
The main question is how to write a test when there are these trivial pixel differences.

The only diffs I got were from the direct plot overlapping the gray part of the right border, while in the saved GIF it doesn't. I am attaching, below, the biggest diff pixel-wise (2 2 12), from frame 54. This overlapping could be a result of @QuLogic's changed transparency handling. However, I don't think it's something users will care about, plus the fact that the border overlaps into the graph is an issue unrelated to his changes.

This is likely an inherent limitation of GIF due to the 256-colour palette. If you histogram the unique colours, you might find that there are 256 combinations already. Otherwise, it's sort of up to the quantizing in Pillow.