I don't think it's technically impossible - all the information that another site needs to properly interpret some activity is in the JSON that's sent. I get the sense that it might be unrealistic to expect Mastodon to make the necessary changes though. It seems more of a political issue than a technical one.
It's partly an issue of keys. Every fediverse actor has a private key and a public key. When my instance sends this to fediverse@lemmy.world, it's signed by my private key, and lemmy.world uses my public key to verify it. When fediverse@lemmy.world sends this comment out, it uses it's own private key to sign it. It can't just re-transmit my comment, because it doesn't have my private key. All it can do is Announce that I've made the comment (and sign the Announce).
Mastodon treats Announces as Boosts, so every post/comment is interpreted as a thing that fediverse@lemmy.world has boosted, so you get all these un-connected posts appearing. I think it's mostly up to Mastodon to remedy.
It works better if a Mastodon actor posts into a Lemmy community, then you get the mix like you imagine. e.g.: https://mastodon.world/@Flash/112095241193510662 (this particular post was crowbarred into Lemmy via !tails@lemmon.website, but it would be the same if the author had done it.)
(sorry to intrude). This has had a few cross-posts on Lemmy. The link that goes to !tails@lemmon.website is attributed to the author of the image, so you can reply to them via that community if you wish.
(sorry to intrude). This has had a few cross-posts on Lemmy - this one technically wasn't, but it seems Lemmy has picked up on it. If you want some ALT text, and to reply to author of this image, the link that goes to !tails@lemmon.website provides that.
The crawler at lemmyverse.net has picked up on an ngrok URL I've used for messing around with Lemmy. I was using it last night to play with PieFed, and saw it make about 20 attempts to read an API that isn't there. I thought - oh, I wonder if that will break it - and sure, enough, there wasn't a 6:00 update from it this morning. Ho hum.
Lemmy doesn't seem to get much recognition in the wider Fediverse - it tends to get bundled as part of 'other apps'. Mastodon is much bigger, so better integration with Lemmy probably gets deprioritised below their own issues and feature requests (e.g. I was reading today that Markdown support is often requested, but the base version still doesn't have it)