It's important to keep in mind the difference between the stock market and the economy.
Economically, AI is a bubble, and lots of startups whose current business model is "UI in front of the OpenAI API" are likely doomed. That's just economic reality - you can't run on investor money forever. Eventually you need actual revenue, and many of these companies aren't generating very much of it.
That being said, most of these companies aren't publicly traded right now, and their demise would currently be unlikely to significantly affect the stock market. Conversely, the publicly traded companies who are currently investing a lot in AI (Google, Apple, Microsoft, etc) aren't dependent on AI, and certainly wouldn't go out of business over it.
The problem with the dotcom bubble was that there were a lot of publicly traded companies that went bankrupt. This wiped out trillions of dollars in value from regular investors. Doesn't matter how much you may irrationally want a bubble to continue - you simply can't stay invested in a company that doesn't exist anymore.
On the other hand, the AI bubble bursting is probably going to cost private equity a lot of money, but not so much regular investors unless/until AI startups (startups dependent on AI for their core business model) start to go public in large numbers.