Yeah, I feel like the emoji system within Lemmy could stand to be improved. There isn't a way in the UI to target an instance and then utilize its emoji. The emoji is retrievable from the API at the https://hexbear.net/api/v3/site endpoint under custom_emojis. Each entry has all the data you need to query the emoji. The interface would need someway to reference other instance's emoji.

    "custom_emojis": [
        {
            "custom_emoji": {
                "id": 68,
                "local_site_id": 1,
                "shortcode": "two-wolves-1",
                "image_url": "https://www.hexbear.net/pictrs/image/da7debc4-92f4-4d35-a27a-1dc2676a3477.png",
                "alt_text": "two-wolves-1",
                "category": "Abolish the Animal Kingdom",
                "published": "2023-06-17T21:15:35.303290Z"
            },
            "keywords": [
                {
                    "custom_emoji_id": 68,
                    "keyword": "inside"
                }
            ]
        },

The base Lemmy UI allows for using the : character to search the local instance's emoji like we have on Hexbear.

Likely, what would need to be added is a way to "federate" or sync the emoji from various instances. I'm sure it would be trivial to write a script that periodically queried a list of instances and pulled their custom_emoji JSON and then import those emojis into your instance. However, what you would really want is a built-in method of allowing two instances to sync their emojis. Federation of emojis is probably not the right way to do it, it would have to be an opt-in process two instances do together, with a way to denoting the origin of the emoji for easy removal.