Yes there is definitely a danger that websites can access your clipboard data.
On desktop browsers, it may be true that this can happen only if you have granted permission or dismissed a warning dialog. But while the permission may have been granted at a particular moment for a particular purpose, the permission lingers beyond that particular moment and ...
I've had a similar case with another app. Google support wasn't of much help identifying the app, and the Marketplace search engine does not support searching by id. But we've been able to find it with a Google search:
The first result is for SketchUp for Schools on Marketplace.