the network can be used to send messages among different parties in even without making an actual txn. but I don't like projects that do that, because doing the same over nostr or some other protocol built for communication has less friction that lightning

if you are thinking of embedding data in the actual txns itself, then whatever is allowed in a Bitcoin txn is allowed on lightning as well, because lightning txns are unbroadcast Bitcoin txns

you can use OP_RETURN to embed data, which will need to re-embeded on each channel update (aka txn), or you can use the much more efficient, yet much more complex RGB (or taro), which just embeds a representation of the actual data on the txns, and the data itself is transferred to the recipient separately