“GreaseMonkey” is only a Firefox add-on. But there are GreaseMonkey-compatible extensions for other browsers, such as Tampermonkey for Chrome (https://chrome.google.com/webstore/detail/tampermonkey/dhdgf...) and GreaseKit for Safari (https://www.macupdate.com/app/mac/20718/greasekit). Thus, GreaseMonkey scripts are automatically cross-browser. Chrome even supports installing GreaseMonkey scripts natively, but only through a convoluted process, for security reasons. The reason people made clones for other browsers is the huge library of existing GreaseMonkey extensions at https://userscripts.org/.
The downside of GreaseMonkey is that the user needs to have a GreaseMonkey-compatible extension installed to run your script. So your extension depends on another extension. It’s also harder to use a build system while writing a GreaseMonkey script, if you want to write it in CoffeeScript or something.