Functionally, I think the Like button serves a very different purpose on Cohost than it does on a lot of other social sites. On Twitter it's basically an "I agree!" button or a "gee that's funny" button or any thing to kind of way to express general approval. But at the end of the day it's just a tiny gesture, and sort of a meaningless one.
I think the consensus on Cohost is that if you like something and want to express that, you can share it (which is meaningful since it exposes it to more people) or you can leave a comment saying that you did like it, what you liked it, or any other kind of discussion/expression of appreciation. The design of the website seems to encourage this through not having the number of likes be visible to the public (or even the poster, unless they really wanna count the number of notifications they've received) to not even including them on comments.
Currently, outside of just using browser bookmarks, Cohost doesn't have a way to bookmark posts you like. However, what it does have is a list in the sidebar titled "posts you've liked." Liking a post still tells the creator that you liked it, however, nobody is able to see your full list of liked posts. Thus, I've found that Cohost's like button is closer in functionality to a "Favorite" button that some websites have, and generally paints a better picture for users of what the button is, that being a private interaction between you and the poster saying "hey, nice work" and a way to quickly jump to those posts again later.
Compared to how other sites do them, I think it's safe to say that Cohost likes have very little in common with traditional "likes" and that a simple name (and maybe icon?) change would communicate that well. Of course, I could be totally wrong about this but that's just how I see it.