I agree with the suggestion that people just don't commit as many things to memory anymore, but that doesn't explain why we still remember certain random numbers from so long ago, while forgetting others. I can instantly recall my ICQ number, but there are plenty of old friends' phone numbers I've forgotten, and those were people I called multiple times a week for years.

Here's a free hypothesis: Maybe it was important to remember your ICQ number. Without it, your message history and contact list was out of reach. In that sense, it fits in the same mental space as a password. What I mean is, there was a cost to losing it. So, you were incentivized to commit it to memory in a way that you weren't with many other numbers.

In contrast, while it would be inconvenient to forget a friend's phone number if you wanted to chat with them, at least you had options. You could generally look them up in the white pages, or call a another friend and get their number, or just ask them when you saw that person again at school the next day.

So, the cost for not remembering a phone number was lower than the cost for not remembering your own ICQ number, and this probably made it mentally stickier.

... Another possibility is a confirmation bias. Maybe we're just not hearing from the 95% of ICQ users who can't remember their number.