Is it possible to use one video type across all browsers?

Since we are using an HTML5 video player with Flash fallback (meaning Flash is utilized if the browser does not support HTML5), you are able to create a single file which will play on all major browser types, including iOS. Our recommendation is to encode your videos in H.264 in an MP4 format.

MediaElementsJS supports most mainstream browsers as well as mobile browsers, however, support for older browsers has been dropped (supports IE11+, Chrome, FF, Safari, Android 4+ and iOS 8+). WordPress will be updating MediaElementsJS to version 4.2.6 and has stated the browsers requirements  here