I have created a simple live stream using Azure Media Services. I can view it from Azure Media Player but I want to be able to embed the video to my personal website using the <Video></Video> HTML tag. I've been looking for a solution but I can't seem to find one. Is this even possible?
The HTML5 tag does not always support playback of HLS or DASH depending on the platform you are on.
https://caniuse.com/?search=hls https://caniuse.com/?search=dash
Sounds like you are trying to playback HLS or DASH from Azure Media Services with just an HTML5 video tag and no javascript player framework. That won't work in most browsers.
Review https://howvideo.works/ for background on how HLS and DASH actually work in the browser - and the reason that you need access to a javascript library and the Media Source Extension feature of HTML5 (not always available in all browsers - see https://caniuse.com/mediasource)