We can associate a one counter finite automaton with a function $f:\Sigma^* \to \mathbb{N} \times \{0,1\}$, where $f(x)=(n,b)$ describes the state where the automaton terminates when fed an input word $x \in \Sigma^*$: $n$ holds the number inside the counter, and $b=1$ iff the state is an accepting state.
Is it decidable to determine whether two of these automata are associated with the same function?
I looked at posts like this one and this one too but I can't find the answer.
Furthermore, since these machines are always aimed to accept/reject a string, is there any natural way to expand them to represent a function? As an example, think of a machine that simply counts the number of times it visited a specific node in the DFA, or even simpler: counts the length of its input word.