But the dot-coms were companies not the technology. The success of the internet itself never faltered, it's just that people scrambled to speculate on companies to capitalise on it, and some terrible or highly risky companies got too much money. It's easier to say Bitcoin itself will succeed than it is to say that blockchain.info or coinbase will be the next google. There's also another dimension to crypto in that there can be an alt-coin bubble in parallel.
So, with the dot-com bubble, pets.com is often used an example of the crash. The pets.com of crypto could be a company (e.g. gox), or it could be an alt coin (e.g. litecoin), but it's not crypto itself. Just as HTTP and SMTP (email) went from strength to strength, so will crypto.
On the otherhand there is a possibility that another crypto overtakes Bitcoin, but this won't happen overnight, and any Bitcoin holders would have plenty of time to diversify towards the alternative before suffering any real loss.
Emphasized for emphasis.
They will likely speak of a "crypto bubble" sometime soon, emphasizing the failure of all other cryptos (and implying a failure of bitcoin as well, most likely just after a peak during a bear market). But indeed bitcoin will be like the internet. It will continue and it won't care what happens around it. And they will surely make reference to the dot com bubble. Bad news sells, good news does not.
And with the millions upon millions invested in bitcoin I doubt anyone will seriously even think about switching to a different crypto. It's not as easy as pressing "delete myspace account" and "welcome to facebook".