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This problem is entirely with Giphy (or your browser?) and not Aseprite, they’re interpolating the image smoothly without also re-interpreting transparency as a full, intterpolatable alpha channel. The result is that the visible pixels become mixed with the pixels that should be transparent, but they no longer are. In other formats, the solution to this is to extrude the pixel colours into the transparent area, but GIF does not support this, so there’s nothing Aseprite can do here.
The good news is if your gif is viewed as a gif at 1x, it should be fine, so long as Giphy doesn’t permanently scale the art.
Something you can try is prescaling your art so that it’s larger than Giphy will ever need to display it. That won’t eliminate smooth scaling artefacts, but it should reduce them significantly.