I guess it depends on what people want the default behavior to be. I couldn't tell you if the share option not cloning tags from the parent post was a deliberate choice or not by the staff, but, in my mind, if nothing else, it should be configurable if this is "opt in" or "opt out" by default (probably opt out at this point, since that's the established precedent).
If you've decided to opt in to automatic tag filling, it could probably simply copy the tags of the original post to get you started, and then you can add, delete and edit as you like from there, and the fastest option would be to simply clone the original post's tags.
That being said, I'd be lying if I said I understood what your included image was conveying, since it simply looks like a different UI for a thing Cohost already has (which is, filling in tags) to me.
Oh, I'm not even thinking about tag cloning (though I think that's a valid point of discussion, for sure!) I'm more thinking about UI options for quick-rechosts (the action that is currently performed by shift-clicking the rechost button).
As of right now, that's a zero-options action -- you shift-click, the post is reposted without any tags or commentary. What I would love is an ability (somehow) to add tags to a repost without having to open up a full page to do so.
Ah, yeah, I didn't even know that was a thing. That makes more sense now.
So essentially: You hover over the share button, you can click it to go to a fully featured share form or hold shift as well to make a popout appear. The popout lets you edit tags for your share without loading a new page. Is that a succinct understanding of your suggestion?
Something like that, yes! I think there's probably a number of ways it could be accomplished, but the goal would be a way to get around the full New Page Option when all i want to do is reblog a picture of a frog and add #frog to it.