I learned about this form of Japanese theater in the late 1990s, from a National Geographic special on Japan, I think. Governor Schwarzenegger famously referred to our state budgeting process as a kabuki dance, maybe around 2008 or so. This is how I learned about the meaning of the term in political jargon. I seem to recall encountering multiple additional instances during the Obama administration, as part of Republican criticism. I have never used the word myself and don't know anybody personally who uses it.
[Later: ] Looks like I was off by a few years:
June 23, 2005
It’s Time to Drop the Curtain on Sacramento’s Bad Theater of Budgeting
Sacramento — Gov. Arnold Schwarzenegger has one thing right. Sacramento’s annual budget drama is “all part of the Kabuki,” he says. “Which means this whole song and dance....
“Everyone is beating on their chest and saying, you know, ‘We are the stronger ones. We’re going to stick to our policies and to our way of doing things,’ and so on. And then eventually you will go and resolve some of those differences.”
In Japan, traditional Kabuki drama is revered. In California, the Sacramento version is reviled.