Not for me. (I suspect the details of this vary between system versions, and they certainly depend on settings chosen in System Preferences / Keyboard / Shortcuts.)
It's the standard behavior. If you do something to modify or override it of course it won't work, but that's like changing the keyboard to Dvorak and saying that pressing the g key gives an i instead so you can't count on getting a g when you press the g key.
It's standard for Cocoa-based apps, I think, but it doesn't appear among the system-wide shortcuts in System Preferences on my Big Sur system, at least. The only shortcuts offered under Input Sources there are to select the previous or next input source (which are set to Cmd-Space, Cmd-Opt-Space).
A command to open the Emoji & Symbols palette is generally at the end of each application's Edit menu, and that's where its shortcut appears. But in current Firefox the item in the Edit menu shows the shortcut as Cmd-Space (not Cmd-Ctl-Space), and it doesn't work for me because the system-wide shortcut takes precedence.
If I disable that shortcut in System Preferences (which may well be the default, particularly if multiple input methods are not enabled), then Cmd-Space does work in Firefox to bring up the Emoji palette -- but note that it's not the standard Cmd-Ctl-Space combination.
Works fine for me in latest Firefox. I've never had it not work in a specific app. Though, sometimes the keyboard crashes and doesn't come back up until a restart (that could be the fault of my Ryzen Hackintosh, though).
