There's quite a few different types of technology (and computers) referenced in the world of Serial Experiments Lain, while some of them are easily matched, some of them aren't. In addition, some of the technologies weren't available at time of production (for example: speech recognition).
The Wired is similar to The Internet and runs on IPv4.
Based on their naming schemes and designs, they're most likely a reference to Apple Inc.
Alice's Navi looks like the bubble iMac G3:
#Lain's original Navi
Lain's original Navi is based off the 20th Anniversary Macintosh:
The Navi systems run C:
#Common Lisp Language:
The Navi systems also run Common Lisp (A dialect of Lisp):
The OS that the Navis run, Copland OS is based on a failed Apple project from two years prior.
The HandiNAVI is similar to the Apple Newton line.
Protocol Seven is Internet Protocol version 7 (drafted in 1993) an RFC that was never implemented in favour of Internet Protocol version 6, whose RFC was drafted in December 1998, after the series had finished.
In Layer 02, Lain gets a chip that lets her interact with the Wired without standard input devices. This is similar to the concept of Virtual Reality, yet is beyond today's technology.
#Unavailable at time of production:
#Navi's speech recognition:
Maybe too late but I'd be willing to chime in and note that Mac OS had speech recognition with assignable actions (Speakable Items) back in Mac OS 9 if not even earlier, and, though a bit limited, it could not only wait for a keypress, but constantly listen for a keyword that designates a command will be said after it.
So, in fact, this was, not as idealistic, but a reality of that time.
And for video chat there was a program called iVisit also in the times of Mac OS 9 which was a bit limited yet could provide some entertainment :-)