Is 2017 MacBook Air SSD compatible with mid 2013 MacBook Air 11"?

This is false: Let $A=K[x]/(x^3)$ and $S$ the unique simple $A$-module. The injective hull is $\pi : S \rightarrow A$ with cokernel of dimension 2 and thus not simple.

Is there an assumption missing? (it would help if you cite the article) For example it is true when $I(S)$ is also projective and $A$ has Loewy length 2.

Let me explain the underlined portion of Lemma 2.2.

Lemma. If $h\colon A\to B$ is a nonzero module homomorphism, then there are a simple module $S$, its injective hull $I_S$, and a map $q\colon B\to I_S$ such that $qh\neq 0$.

Apply this in the proof with $A=P_{S}$, $B=I_{S_r}/S_r$, and $h=pvf$.

Proof of Lemma. Since $h\neq 0$, there is some $a\in A$ such that $b:=h(a)$ is not zero. Let $C\leq B$ be a submodule of $B$ maximal for $b\notin C$. There is a least submodule of $B$ properly containing $C$, and it is $C^*=\langle C\cup \{b\}\rangle$. The natural map $\nu\colon B\to B/C$ is a surjective homomorphism onto a module $B/C$ that is an essential extension of a simple module $S:=C^*/C$. The module $B/C$ embeds into the injective hull $I_S$ of $S$; let this embedding be written $\iota\colon B/C\to I_S$. Now define $q:=\iota \nu\colon B\to I_S$. The composition $qh$ is not zero since $qh(a)=q(b)=\iota\nu(b)\neq 0$, since $\nu(b)\neq 0$ and $\iota$ is an embedding. $\Box$

Why do SSH requires but doesn't care the hostname?

Because the control sockets SHOULD be based on the hostname you are connecting to, as described in the manual page for ssh_config:

ControlPath

Specify the path to the control socket used for connection sharing as described in the ControlMaster section above or the string none to disable connection sharing. Arguments to ControlPath may use the tilde syntax to refer to a user's home directory or the tokens described in the TOKENS section. It is recommended that any ControlPath used for opportunistic connection sharing include at least %h, %p, and %r (or alternatively %C) and be placed in a directory that is not writable by other users.* This ensures that shared connections are uniquely identified.

If you do not follow these recommendations and try to connect to different host with the same socket, ssh finds the existing socket and uses it regardless the written hostname.

To the edited question:

Would I ever have a problem if I use this command as follows ...

ssh -S /path/to/my-first-host.socket example.com .... # expect connecting first host
ssh -S /path/to/my-second-host.socket example.com .... # expect connecting second host
ssh -S /path/to/my-third-host.socket example.com .... # expect connecting third host

Yes. It is not enforced/used now, but you are clearly using undocumented/undefined behavior, which can change at some future release without any notice.