So, after lots of searching, I came to know why many Emojis by default are not colored in Edge on Windows 10. Seems like Segoe UI Emoji font needs to be applied for the emojis to be rendered in color, otherwise they use black outlines format.
I changed my standard font from Segoe UI to Segoe UI Emoji in Settings, and now all emojis render how they are supposed to.
Emojis when Segoe UI is set as defaultEmojis when Segoe UI Emoji is set as default
I would request the team to apply Segoe UI Emoji by default for everyone so that they can take advantage of the same. I also came across the finding that Firefox does this by default, but Microsoft Edge and Google Chrome does not. Please make the changes for a better experience using the emojis.
Tagging @Deleted and @MissyQ for conveying my feedback to the team.