Yeah, people talk about "functional languages" as if they're a unified thing. IMO the differences between the strongly typed ML/Haskell tradition and the Lisp tradition are as big as the differences between either and "OO languages" or "imperative languages", but that's not the way it's usually presented.

That's nonsense. Lisp in the functional style vs. Haskell is two different implementations of what are at their essence the same ideas. However, lisp is multi-paradigm, which complicates the comparison somewhat...

What are the essential features that they share?

They're both primarily based on lambda calculus. Although haskell on typed lambda calculus. They both emphasize purity over mutations. They both are designed to make higher-orderisms idiomatic.

But the point isn't so much the features they share. I'd be the first to admit that Lisp and Haskell are very different. But functional programming in both is very much based on the same ideas. Claiming that they're as different as OO vs imperative is like saying that washing dishes by hand is as different to using a dishwasher as football is to baseball.

"They both allow you to pass functions as arguments to other functions."

" Is it because Lisp is seen as the birthplace of functional languages (because of point 2)?"

That was my feeling. Some tenuous connection that leads to meaningless comparisons. They have such different styles, cultures, and attributes that direct comparisons don't even make sense. People might technically be able to do something that's unnatural in one in the other easily. And vice versa. Question: would any idiomatic user of either really want to or waste time doing it that way? Probably not...

Both language communities are enamored of manipulating and interpreting various flavors of rose tree.