The size limit for animated GIF emojis is 256 kb, and the maximum file size for profile pictures is 8 MB. You can upload a larger image, but Discord will automatically crop and resize it to the appropriate size. Don't get confused between what a Discord avatar is and a profile picture; they are the same thing. It is also called a Discord PFP in some cases.
When you're setting up your server, you can also add a Discord banner background image and a Discord server invitation presentation image. Discord gives you plenty of screen space for these images, allowing you to customize your server. The banner background image can be up to 960 pixels wide by 540 pixels high, and the server invitation welcome image can be 1920 pixels wide to 1080 pixels high. Discord users who have a free account can only send files up to eight MB.
However, this limitation does not include GIFs in the Discord text box. If you upload an image, video, or other file format, only the smallest files will appear. There's also a premium paid version of Discord called Discord Nitro, which offers all sorts of cool boosters like custom emojis, animated avatars, and more. When your GIF is in Tenor, you can send it to everyone since it's not uploading your file through Discord. WhatsApp does not send the message as a GIF image; in fact, the message is sent as a video itself and since the video is short, it actually plays it in a loop on the screen, which looks like a GIF animation without sound.
The upper limit of the file size for GIF avatars is 50 MB. If your search for GIFs on Instagram doesn't work, it could mean that your application is not updated to the latest version or that it's due to an error. In addition, the typical file types you can use are JPG, PNG, and GIF for Discord animated profile images. Another thing to keep in mind is that Discord can't crop GIF avatars to a square like it does (if I remember correctly) for normal images. If you use a messaging program like WhatsApp (which now supports GIFs and videos on Android), make sure that both users use the same version of the application.