That's interesting, as a native American English speaker approaching 40, I have never heard anyone say "go for a bike". Neither native speakers nor people whose first language is not English.

Is this common in some part of the country, or outside the US?

Yeah, from western Canada, i'd go biking, or go for a run, but I don't think I've ever said, I'm going to go for a bike. I'd say "I'm going for a bike ride".

I've never heard anyone say it in the UK either. I think the OP's reasoning is off. Just because 'bike' can be used as a verb doesn't mean that it will get used in that way in all situations.

My knowledge of grammar isn't great, but isn't bike(-ride) working as a noun here? I can use it as a verb and say "I biked here this morning" or "it's faster if you bike across the bridge". But I've never heard "I'm going for a bike", or going for a "20 mile bike". Bike ride is what sounds natural in those sentences.

And yet you can "go for a run/jog/walk/swim". English is weird. I guess it's because bike as a noun already means the bicycle itself, while there is no object named "run/jog/swim".

Saying "I'm going for a bike" would be the equivalent for "I'm going for a sneakers" (for a run).. its the action of riding that makes sense.

I thought the author was reaching for a questionable "Tale of Two Cities" Dickens pun :\

This is true in Dutch to a larger degree than English. It seems several English people learning Dutch on duolingo are confused that there is a verb that means "bike". https://www.duolingo.com/comment/3786991 for instance.

Still, you'd never say '10 fietsen/bikes'.

You'd in Dutch, just as well as English, say '10 fietsritten/bike rides'.