Is it really the case that Mozilla Firefox is a modern day version of Netscape Navigator?
Yes, it’s true, as explained in Wikipedia’s entries for the Mozilla application suite and Firefox. More accurately, Firefox is a descendant of Netscape Navigator; most of Firefox has been rewritten in one way or another since the days of Navigator.
Most of Netscape Navigator (or rather, Communicator) was released as open source in 1998; this was then extensively rewritten and eventually emerged as the Mozilla application suite. Some of Netscape’s engineers started developing a subset of the suite to produce a nimble browser, and that eventually became Firefox (after conflicts over its previous names, Phoenix and Firebird).