With the POSIX API, you can duplicate the socket file descriptor, but calling shutdown() to close one direction of a file descriptor will actually close that direction on all file descriptors... unfortunate.
As with so many things in computing the abstraction leaks a bit when you get deeper into it but as a first intuition for beginners I think you could do way worse than "a socket is a bidirectional pipe to another computer".
I'm intrigued by your idea of making 2 file descriptors for each socket btw. Why would it be such an improvement in your opinion?
TCP is complicated. But "stream based sockets" aren't "just" TCP (ex: Unix domain sockets fixes all of the "glitches" of TCP).
The important bit of "sockets", in general, is the concept of:
1. "Accepting" file descriptor (creates new file descriptors whenever someone is "connecting" to you).
2. Multiplexing and "juggling" those descriptors.
TCP fits this model, though with some low level quirks. Unix Domain Sockets are arguably the place to start, but most beginner programmers wouldn't find that useful (they really want internet connections).
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Its difficult to make a "server" off of one pipe. Maybe two pipes (first pipe accepts requests and divvys out new pipes to work off of). But no matter how you do it, its wonky.
A proper server needs that "accept" abstraction, which is completely missing from pipes.
You can receive file descriptors over a UNIX socket, but yeah. That said I don't see why a file descriptor is a good abstraction for a listening socket in the first place when you don't read() or write() to it.
Pipes guarantee delivery if both sides agree, tcp doesn’t (but it tries hard asynchronously), that’s where the difference starts.
A socket API that gave you two unidirectional fd's for a TCP socket would have been a massive improvement.