The "formal" part comes from "formal system", which is essentially the use of rules of inference based on an initial set of presuppositions, with an exact/mathematical approach to "truth".

The idea of "informal logic" is something some people have wanted to put forward, but it's a much younger and not as well defined term, and whether it should really be considered "logic" is something that has criticisms: https://en.wikipedia.org/wiki/Informal_logic#Criticisms

Perhaps in the end all logic is formal logic, the problem is that we know from Gödel's incompleteness theorems that not everything is computable and translatable into a formal system.. to solve this you'd have to model the statements into a Gödel/Lobian machine (at which point you are essentially building an AI).