I should have been clearer with the IBAN point. In particular I was not talking about people outside of the banking system. I'll respond here to most replies I got so far.
What I meant is that you need the IBAN of the person you want to send money to (of course you need to have an IBAN, as in, you need a bank account for your MOB to be actually worth anything). Assuming Alice and Bob are contacts on Signal, and Alice wants to send money to Bob, Alice still has to ask for Bob's IBAN. We can safely assume that Bob does not know his IBAN by heart, and has to use his ebanking app (for instance).
You already have some kind of identifier for Bob, Signal would allow you to reuse it. Some ebanking apps already allow you to do that, but I think in most cases both have to be clients at the same bank (?)
The idea is that you have less friction. The chat would be something like:
> B: can you send me 5 USD
> A: sure, here it is
> A sent you N MOB, equivalent to 5 USD.
Of course the UX is still far from being there (volatility is the biggest problem I think).