That browser may have been built using open source Firefox 3.0 based code 6 or 7 years ago, but it wasn't built by Mozilla - it was a 3rd Party build which isn't supported by Mozilla.

I think that would be a question for Sony, who may have contracted to have a vendor built that browser from source code. https://us.en.kb.sony.com/app/answers/list/kw/Sony%20Vaio%20Splashtop%20browser/c/65,66/

BTW, I have a 'netbook' device from that era (2007 - 2008) that came with a custom built version of Linux and an older version of Firefox. I was able to locate a 'slim' build of Linux so that I was able to install a newer Linux OS that had a much newer version of Firefox. Not as easy to install a newer version of Firefox on an older Linux OS, as it is with Windows and in some cases MacOSX.