Oh it's a lot more convoluted than that.
For all intents and purposes when you think of the AT&T monopoly of yore, actually Verizon is more of that than the current incarnation of AT&T that is entertaining us today with this legal battle.
First, AT&T was divested. The monopoly part became mini-monopolies - the Baby Bells. They were still almost exclusively the only show in town for what they did (local telephony). AT&T actually had to compete at that point, on several fronts. Long Distance became a highly competitive arena over time. And the part that made telephony infrastructure equipment could no longer simply dictate to the local phone companies what they were gonna buy.
The first wave of Wireless in the US was a mandated duopoly. Each area got two licenses for wireless service providers. The "B" band went to the established phone company while the "A" band was up for grabs. The "B" side was often termed the "wireline" side because they were established companies already. Gradually, a large chunk of the upstart "A" side companies coalesced into McCaw. Before the "B" side companies started merging, McCaw was actually bigger than most.
Eventually AT&T bought McCaw and became or created AT&T Wireless.
The game changed with lots more licenses and more players.
SBC bought up Ameritech, then AT&T and then changed it's name to AT&T.
In all of that, if you restrict your view to the Wireless stuff Verizon is much more directly a descendant of the Baby Bells.