When running under GNOME 3, Signal does not use system emoji set, nor does it use the system emoji picker.
Signal's emoji set and picker is on the left, GNOME's on the right.

Signal by design currently uses a consistent emoji set across all platforms.
I personally think that should be up to the user! Sure, you can use those (in my opinion ugly) apple emojis by default, but just like with the android app, the user should have at least the choice to use system emojis.
Prior discussion: https://github.com/signalapp/Signal-Desktop/issues/975
For anyone finding this in the future and beeing of the same opinion: Here is a fork that simply replaces the apple emoji datasource with the google variant: https://github.com/Skycoder42/Signal-Desktop-Noto. It is also available as AUR-repo "signal-desktop-noto"
The Android app allows you to use the system emoji. The desktop app should allow you to do the same for feature parity. At the very least, the desktop app should allow you to select the Google emojis. The Apple ones are extremely ugly. Telegram has this feature for a while now.
@kenpowers-signal Please make it available to install other set of emoji's and preferably make available noto version of these.
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The Android app allows you to use the system emoji. The desktop app should allow you to do the same for feature parity. At the very least, the desktop app should allow you to select the Google emojis. The Apple ones are extremely ugly. Telegram has this feature for a while now.