An important part of mathematics is indeed formalizing such objects. Mathematicians go further, question these assumptions, and abstract from them. For example, removing constraints on fields gives us rings, groups, semigroups, and categories, and exploring what can still be said about them.

But I'm sure pure mathematicians would also be totally fine studying randomly generated rulesets. The only restriction is that they should not lead to immediately obvious contradictions, since those would probably be kind of boring!