How can every regex be accepted by an infinite number of finite automata?
There are at least two ways of doing this.

The first is kind of trivial: there's no requirement that all states of an automaton be accessible from the start state. So, take your favourite automaton that accepts some language, and then add as many new states as you want, with arbitrary connections between them, as long as they're not connected to the original automaton.

A second way keeps the automaton connected. I'll give an informal description; I wrote a more formal one in response to another question. Take your favourite automaton and make $n$&nbsp;copies of every state. Modify the transition function so that if, in the original automaton, $\delta(q,a)=q'$ then, in the new automaton, reading&nbsp;$a$ in the $i$th copy of state&nbsp;$q$ takes you to the $(i+1)$st copy of&nbsp;$q'$ (with addition modulo&nbsp;$n$). The start state is the first copy of the original start state; every copy of an accepting state is accepting. This isn't guaranteed to keep the automaton connected but you can delete the disconnected states and you are guaranteed to end up with an automaton with at least&nbsp;$n$ states.