Is “Open kimono with somebody” a popular English phrase?
To open the kimono to someone
is a bit of business argot. It is figurative, meaning to give away or be open about a few secrets to entice the buyer and encourage a deal to go forward. In a pitch to an possible investor, the inventor doesn't want to give a way the 'secret sauce' (the important idea behind the invention because the investor might still the idea. But the inventor has to show -something- that will entice them to prove that there is something there that is worthwhile. So the inventor will give some few details that will be convincing.
The metaphor is in reference to a geisha flashing open her robe to a client to engage his interest. Americans (where the phrase presumably originated) are only familiar with the kimono as something worn by geishas and have no idea of the associations (or lack thereof) it has natively.
'Open kimono' is a derivative of this and has come to mean 'to show the innerworkings', less a sneak peak but rather full access. In the OP phrase we are totally open kimono with regulators, the speaker was just trying to get across that they won't be hiding anything from the regulators.
The grammar of 'open kimono' is a bit informal by converting a 'verb noun' pattern to an 'adj-noun' (essentially nouning a verb). Using it as an adjective is even more familiar:
we are X with you
where X is an adjective is not formal or standard English, but is very colloquial (that is, not good for writing, but works in speech). It is not common in general but is recognized in the business world.
Open the kimono or flash the kimono is reasonably common in US financial use – meaning to show everything.
To Americans and most westerners, a kimono is a robe (BE dressing gown) so the assumption is that you are naked underneath.
I don't think it's very common among 'normal' people.
I think it is a widely-understood phrase in business, but perhaps a little pretentious.
You certainly wouldn't translate 'kimono' into its English equivalent.
When trying to understand what a word means in English, using a Japanese/English dictionary may not be the best approach. Instead, use an English/English dictionary.
For example, "sake" in English means rice wine (nihonshu), whereas in Japanese it can mean alcoholic beverages in general.
As mentioned by Jim, "kimono" seems to have a different meaning from Japanese's "kimono" (which would literally translate as ki + mono, "wearing thing").
Some westerners believe that women don't wear any underwear underneath a kimono. For example, the author of TV Tropes' page on kimono fanservice. I don't know if this was believed by the users of this idiom, though.
It's quite common in my industry, software development. Companies, especially US based ones such as Microsoft, like to use it to state that they are now being very open about their processes, software and internal systems.