There was certainly a vast gold rush around the internet with massive attention from almost everybody in journalism or in business and almost every retail investor -- at least in the US (where I was) -- but it differed from the AI hype in that there was no vision of utopia or at least not any that resonated with more than a very small fraction of the population.
The closest the internet got to becoming the focus of hopes for a great transformation of society (or the focus of an ideology or comprehensive narrative) that I saw was a conversation started by John Perry Barlow's "A Declaration of the Independence of Cyberspace" (published Feb 1996) and the crypto enthusiasts. Both conversations consisted mostly of libertarians (with the crypto conversation also having anarchists), which as everyone knows only ever made up 4 or 5% of the US population (but a much larger fraction of internet users in the 1990s).
Also, Teilhard de Chardin's books (written in the 1950s and 1960s) inspired people with a grand vision of the human future and in particular predicted the creation of a "noösphere" which was much like the global internet, so Teilhardists wrote some about that. But the Teilhardists were never more a tiny tiny group, much smaller even than the aforementioned crypto enthusiasts. Almost not worth mentioning.
Was there a general sense that something singularity-like was approaching?
No. Although millions of words were written during the 1990s about the singularity (referring to it using that word specifically IIRC), even then it was always AI that was expected to bring about a utopia, never the internet. And the discussion was restricted to a few mailing lists (the extropians list, Yudkowsky's SL4 list and ISTR a transhumanism list), one or two books (e.g., Great Mambo Chicken and the Transhuman Condition), Omni magazine and maybe Wired magazine.