My version of Internet Explorer uses the following user agent.
User-Agent: Mozilla/4.0 (compatible; MSIE 8.0; Windows NT 6.1; blah blah blah
Theoretically, can Mozilla Foundation sue Microsoft for use of their trademark?
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Regarding theoretical possibility of lawsuits, Super User cannot provide (valid) legal opinions. However, as to why IE identifies itself as Mozilla in the user-agent string, see this (somewhat tongue-in-cheek) blog post for the nitty gritty.
Essentially, the Mozilla Foundation didn't exist (as we know it) until the beginning of the 21st Century. The name Mozilla was taken from the original name of the browser which would eventually become Netscape Navigator. IE supported Navigator's special additional features, but wasn't actually Navigator, and thus identified itself in this way to flag that it was compatible with the Mozilla family. At the time, using the browser's user-agent string was a common way to do compatibility testing. (Now there are much more advanced ways which revolve typically around detecting the actual DOM/scripting capabilities of the browser.)
When Netscape Navigator was killed, it was rebirthed as the Gecko HTML rendering engine and the Mozilla browser. The Firefox branding came about a bit later, but several other Gecko-powered browsers existed at one point. (I distinctly remember using early versions of Camino on the Mac, I believe prior to Firefox's existence on Mac.)
No, mainly because of this thing called "for purposes of interoperability"
(Warning, sarcasm ahead.)
In ancient internet times web sites began to check user agents. They would throw a shoe and display bad HTML, or scary warning messages to the user if the browser wasn't "Mozilla" because obviously a browser that wasn't Mozilla couldn't display their advanced content.
So IE had to call itself Mozilla.
And then browsers that didn't say they were MSIE or Netscape couldn't get advanced content from sites, because they were deemed "too old" but really they were Opera, or Konqueror, or Safari, or Chrome...etc.
And so began, as John Rudy's answer stated, a long line of "I'm really X, but so everyone else will show me their content for Y, I'm going to say I'm Y (but X wearing a mask)"
And so basically, user agent sniffing is useless.