Is it consistent with $\mathsf{ZF}$ (without $\mathsf{AC}$) that there is an infinite set $X$ and a subset $S\subseteq\mathcal P(X)$ of the same cardinality as $\mathcal P(X)$ with the property that whenever $s\neq t\in S$ we have $|s\cap t| \leq 1$?

Note that it is consistent that there is such a pair of sets $X,S$, with $|S|>|X|$, but no such $X$ can be well-orderable. See here and here.

Is it consistent with $\mathsf{ZF}$ (without $\mathsf{AC}$) that there is an infinite set $X$ and a subset $S\subseteq\mathcal P(X)$ of the same cardinality as $\mathcal P(X)$ with the property that whenever $s\neq t\in S$ we have $|s\cap t| \leq 1$?

Note that it is consistent that there is such a pair of sets $X,S$, with $|S|>|X|$, but no such $X$ can be well-orderable. See here and here.