Internet as it formed at the brink of millenium is a fluke. I remember reading a tongue-in-cheek conspiracy article which argued that even if in the field of mathematic research we had two incompatible standards (Mathematica and Matlab at that time), it's improbable that a single earth-encompassing standard for communications would emerge.
The zeitgeist, by the way, (surveillance, national three-strikes like laws, etc) is for the Internet to be torn onto semi-walled national webs, so value it while it lasts. Not that the most internet users will notice, as the internet for them means their social network site of choice.
That aside, there are several variants you may want to consider.
Imagine an internet without anonymity at all. You go in under your realname, you need to have your local authorities agreement to create a website (like what you need when you're opening a real-life establishment). Imagine South Korea's internet, just slightly exaggerated. This will make a drastically different web from what we know.
For the cheap way out you might describe some oppressive government. However, realname policy worked quite fine in FidoNet, without any government oversight. Your aliens may hyperreact (from our point of view) to any slights (it's in their culture), so to curb any violence the disclosure of personally identifiable info is mandated by benevolent authorities or cultural consensus.
Sure, this approach may not sit well with someone, but the practice of selling phones locked in for SIMs of a particular cellular operator also seems outlandish to plenty of non-US people.
You may have the world-wide network operating on different principles. Imagine a world-wide network based on FTN technology. In fact, just dig into 90s, you'll see it. You have the throughput way lower, with all the synchs and mail-exchange times, your mail may take three hours to get to the other side of the planet, but it worked and it will work.
You may have different, hardly compatible networks developed in different state-like entities. France had its own quite robust network Minitel in 80s (though in the beginning of 90s it was outdated by internet). You just need a strong not-invented-here bias for independent networks to survive.
The internet won't be the way we perceive it now if only a limited number of people could access it (think Eternal September):
- If you're cheap, again, you may go for an oppressive government repressing freedom-loving rubber-mask humans.
- For a moderate variant let's imagine that access terminals just cost too much (think of ARPANet).
- For a truly alien variant your extraterresterials may be just not that socializing. Imagine a civilization of city-states where people are just as interested in talking with other states as you are interested in delving into a zoning debate in some Indonesian province. Sure, you can, but what for? So, only a small share of population will use such a network, scientists, diplomats, xenophiliacs, this sort of people. They will shape their web in a very alien way.