Not always; that's app-specific, not a system-wide setting.

On macOS I use Rocket [0]. It's not perfect but it's the best I've used and it does make finding the emoji you want pretty easy. You can add custom alias/shortcuts for the emojis and you can also use it to insert images/gifs from the searcher.

[0] https://matthewpalmer.net/rocket/

The globe button on newer Macs can be retasked to call up the emoji input panel (I guess any button could be before with custom keyboard shortcuts but now it's a simple dropdown in keyboard preferences.)

Holly mothballs. I didn't know this trick. Works in Slack as well.