I can give you a hypernym for bar and nightclub: they're both drinking establishments.
It's not one word though, it's two. And it's not snappy and informal.
I'm not sure on the use of bar for both bar and nightclub. I (as a native British English speaker) don't tend to use the word bar much at all, I tend (or rather, tended) to go either a pub or a club. A bar is something I found when I went on holiday to Europe.
As you say, somewhere with a proper dancefloor isn't a bar, it's a club. People do make a distinction between a bar and a club, see e.g. the lyrics to Shape of You by Ed Sheeran:
A club isn't the best place to find a lover, so the bar is where I go
All that said, the wikipedia page for bar says:
A bar (also known as a saloon or a tavern or sometimes as a pub or club
and the page for club says:
A nightclub, music club or club, is an entertainment venue and bar that usually operates late into the night
So clearly some people do think that a club is a type of bar, so using bar for both would be acceptable.