I would say exactly the opposite, frankly.
With the internet, there was a clear value proposition for the vast majority of use cases. Even if some of the specific businesses were poorly-conceived or overly optimistic, the underlying technology was very obviously a) growing organically, b) going to be something everyone used & wanted, and c) a commodity.
All three of those parts are vital for a massive boom like that.
Generative AI is growing some, yes, but a lot of the growth is being pushed by the companies creating or otherwise massively invested in gen-AI. And yes, many people try out ChatGPT's webapp, but that's mostly a gimmick—and frankly, many of the cases where people are attempting to use it for more are fairly awful cautionary tales (eg, the people trying to use it as a therapist, and instead getting a cheerleader that confirms their worst impulses).
Gen-AI may be useful to some people, but it's not going to be a central feature of most people's lives—at least not in the forms it exists in today, or what can be clearly extrapolated from them. Yes, it can help some with coding—with mixed results—but not everyone's a programmer. Not everyone's even an office worker. The internet has obvious useful applications for a plumber or a lawyer; if I hired one of those and they said they were using generative AI to help them in their work, I'd fire them instantly. There are already a bunch of (both amusing and harrowing) stories of lawyers getting reamed out in court for using gen-AI to help them write legal filings.
OpenAI may or may not have a robust moat—I've seen people arguing both ways; personally I suspect lean slightly toward the "not" side—but generative AI as a whole is not something that's an interchangeable commodity the way internet access, or even hosting, is. First of all, in order to use the models that are touted as being advanced enough to actually look like more than spicy autocorrect, you need a serious GPU farm. Second of all, AFAIK, those models are being kept private by the big players like Google and OpenAI. That means that if you build your business on generative AI, unless you're able to both fork out for a massive hardware investment and do your own training to match what the big boys are already doing, you're going to be 100% dependent on another specific for-profit company for your entire business model. That's not a sound business decision, especially during this time where both the technology and the legal aspect of generative AI are still so much in flux.
Generative AI may be here to stay, but it's not going to take over the world the way the internet did.