Kate display some emoticons, but not all of them, incorrectly.
For instance U+1F3EB SCHOOL is displayed as
Changing the font doesn’t fix the problem
Looks like a bad font configuration or missing font.
I recently faced a similar issue but for the braille characters used by some programs as indeterminate loading animation and this thread in the Arch Linux forum helped me to understand why it was happening and how to solve it. [Solved] Konsole using less readable braille font for graphs / Newbie Corner / Arch Linux Forums
Please share the output of fc-list :charset=1F3EB and check if the emoji is correctly displayed when you run FC_DEBUG=4 pango-view -t "🏫"
On my system the first command outputs
/usr/share/fonts/noto/NotoColorEmoji.ttf: Noto Color Emoji:style=Regular
If yours looks different or doesn’t have NotoColorEmoji font installed please install it and reboot to see if that fixes it. If it doesn’t you may need to somehow change your fontconfig
Perhaps this is additionally necessary for the NotoColorEmoji font:
https://community.kde.org/Distributions/Packaging_Recommendations#Fontconfig_configuration