I think they mean consensus about which axioms to accept. There could be competing schools of mathematics along those lines, I think there just happens not to be.
Don’t all roads lead to Rome? It’s like an engineer deciding to use SI or Imperial units, people have their opinions but at the end of the day both work.
In theoretical maths, yes.
In the same way some heterodox political philosophies may result in new moral systems, but it doesn’t really matter that these moral systems are different to the status quo, until someone uses it as an ideology for their revolution.
It’s probably good to have a variety of theory to choose from, except when your different theories result in different practical outcomes.
In mathematics you might compare groups that work with Vs without the Axiom of Choice, you can can have one group satisfied by an existence proof that asserts X exists otherwise a contradiction must exist and another group that only accept existence proofs that demonstrate a means to construct an example of X.
The habeas corpus divide :-)