Marking this as confidential, since I'm not sure how critical it is to keep nightly, alpha and release looking the same. But this was the aim of #42319 (closed), but there is a work-around.
and use computed style on #rightBox and #trademark to determine the difference. This works because the CSS is contentaccessible, which means it can be used for anything that isn't a document, including as a stylesheet.
Note this only distinguishes between the stable release and nightly/alpha, because they have different rules for only these two elements.
Technically, alpha has another rule for #rightBox:-moz-locale-dir(rtl) that nightly does not. However, I don't know of a way to access this rule. I think -moz-locale-dir never matches for HTML (rather than XUL), and I don't think it is possible to read the cross-origin CSSStyleSheet.cssRules. But maybe there is some other way.
I guess the solution would be to either:
Use the same CSS for each platform.
Make this CSS identical/empty on all platforms, but add the differences in some other non contentaccessible package. I.e. somewhere other than chrome://branding/content/. We can then do a similar thing with about-wordmark.svg to avoid the padding and right-to-left corrections.
Make all of chrome://branding/content/ non contentaccessible. I asked in the firefox developer chat if there is a good reason not to do this.