At least it does work with Microsoft Edge 94 - when I open mobile Facebook website to write a comment or reply to a message, and then press Windows+Period to open emoji panel and navigate it then press Enter on an emoji, the selected emoji is entered. NVDA also announces selected emojis as I navigate through the emoji panel.
Does this problem happen with Narrator, too? If not, tyhat's something we may need to look into - please ping me next Monday if you still need my assistance (I'm out of action this week in NVDA world due to graduate school midterm this week).