Connecting a socket in timeout mode to a server with full listen backlog raises BlockingIOError

Bug report

Bug description:

I'm working on a Debian bookworm system. Please consider this executable script ms:

#!/usr/bin/python3 -i

from pathlib import Path
import socket

SOCK = 'stream.sock'
s = socket.socket(socket.AF_UNIX, socket.SOCK_STREAM)
Path(SOCK).unlink(missing_ok=True)
s.bind(SOCK)
s.listen(0)

and this one mc:

#!/usr/bin/python3

import socket

SOCK = 'stream.sock'
s = socket.socket(socket.AF_UNIX, socket.SOCK_STREAM)
s.settimeout(3)
s.connect(SOCK)

Let's start ./ms, and once it reaches the Python prompt, run ./mc twice in another terminal:

$ ./mc
$ ./mc
Traceback (most recent call last):
  File "/home/wferi/block/./mc", line 8, in <module>
    s.connect(SOCK)
BlockingIOError: [Errno 11] Resource temporarily unavailable

The first[^1] invocation runs to completion all right, but all succeeding ones immediately fail with BlockingIOError, as if the connecting socket was in non-blocking mode, not in timeout mode. I understand that the documentation[^2] notes

at the operating system level, sockets in timeout mode are internally set in non-blocking mode

so that must be where this comes from, but such an implementation detail shouldn't leak out of the abstraction, as the documentation also states that

A socket object can be in one of three modes: blocking, non-blocking, or timeout.

and

exception BlockingIOError: Raised when an operation would block on an object (e.g. socket) set for non-blocking operation. [^3]

[^1]: In general the first backlog+1, but I set the listen backlog to 0 in the code of ms above.
[^2]: https://docs.python.org/3/library/socket.html#socket-timeouts
[^3]: https://docs.python.org/3/library/exceptions.html

CPython versions tested on:

3.11

Operating systems tested on:

Linux

Linked PRs

• gh-134688

Reproduce the behavior with the ms and mc scripts in the issue, using a Unix socket with listen(0) and repeated timed connections. Read the linked socket timeout and exception documentation, then inspect the relevant CPython socket implementation and linked PR gh-134688. Done means the documented timeout abstraction is preserved when the listen backlog is full, with regression coverage for the reproduced case.