The versions of Firefox and Chromium in the AppCenter are packaged by the Ubuntu developers. I would be surprised if the elementary team packaged their own custom versions of these applications in the future; their focus seems to be on the included Epiphany browser.
If you'd like elementary to show up in your user agent string, you can use Epiphany as Maxim suggested, or use a user agent switcher plugin. User Agent Switcher for Firefox and User-Agent Switcher for Chrome/Chromium should do the trick.
All that said, many analytics packages summarize things down to Windows, macOS, and Linux. As a web developer, I keep an eye on analyltics for Linux but don't typically break things down by distro. Linux as a whole represents about 2-3 percent of traffic.
If you'd like to encourage web developers to improve support for Linux browsers, just browsing with the default user agent string should do the trick! :)