It is not version specific. It is linked to any program named “sketchup.exe”.
Here is a zip archive with 4 .reg files for setting versions. (They likely require administrative priviliges to change the registry. Windows UAC may first prompt you for confirmation, or ask for an admin password if you are on a standard user account.)
SketchUp_WebDialog_BrowserEmulation_regfiles.zip (1.5 KB)
Note: .reg files are by default associated with the regedit executable. When you double-click a .reg file, regedit will prompt you for confirmation to run the file and make changes to the registry. These files are plain text and can be viewed in any good code editor (of which plain-jane Notepad is not.)
DISCLAIMER: You take responsibility for changing the browser emulation setting. The Trimble SketchUp team is working on updates to Dynamic Components, etc. to try to work around the emulation issues. They may want it set to a specific setting when all the pieces fall into place later.
So consider any change you make temporary.