Can math be subjective?

There's plenty of room for subjective opinion in mathematics. It usually doesn't concern questions of the form Is this true? since we have a good consensus how to recognize an acceptable proof and which assumptions for such a proof you need to state explicitly.

As soon as we move onwards to Is this useful? and Is this interesting?, or even Is this likely to work?, subjectivity hits us in full force. Even in pure mathematics, it's easy to choose a set of axioms and derive consequences from them, but if you want anyone to spend time reading your work, you need to tackle the subjective questions and have an explanation why what you're doing is either useful or interesting, or preferably both.

In applied math, these questions are accompanied by Is this the best way to model such-and-such real-world problem? -- where "best" again comes down to usefulness (does the model answer questions we need to have answered?) and interest (does the model give us any insight about the situation we wouldn't have without it?).

The subjective questions are important in research, but can also arise at more elementary level. The high-school teacher who chooses to devote several lessons to presenting Cardano's method for solving the generic third-degree equation will certainly have to answer his students' questions why this is useful or interesting. Perhaps he has an answer. Perhaps he has an answer that the students don't agree with. In that case, he cannot look for a deductive argument concluding that Cardano's formula is interesting -- he'll have to appeal to emotions, curiosity, all of those fluffy touchy-feely considerations that we need to use to tackle subjective questions.

Every single statement, question, and claim in mathematics is subjective because they are always based on a set of axioms, which are arbitrary, and are picked to observe their consequences.

However, once you phrase the claim in the form of an implication, (such as: "if [the axioms of Euclidean geometry], then [the Pythagorean theorem]") then you have an objective truth. This is assumed to be the meaning when any mathematician states a theorem - we understand what axiomatic framework they are working in and understand that their claim is contingent on those axioms.

Given a particular axiom system, three of the possible results for a mathematical claim are:

We prove the claim true. [Ex: The Pythagorean theorem]

We prove the claim false. [Ex: "The integers under multiplication form a group"]

We prove that the claim is independent of our axiomatic system. [Ex: the continuum hypothesis].

(see Mario Carneiro's comment for other possibilities).

There are no claims that can be subjective if we take for granted that we are working in an axiomatic system. Some people might argue that the continuum hypothesis is "subjective" under ZFC, but I prefer to think of it as simply having no truth value.

There are a few more points of subjectivity that are related to, but slightgly different from the sometimes subjective choice of axiom system: I am talking about definitions of some standard objects. For example, people may have different "opinions" whether or not $0\in\mathbb N$. Or whether they accept an answer to a question asking for an explicit solution only if it is elementary (a combinations of polynomials, trigonmetrics, esponential, logarithm) or if they would also accept something involving the Lambert $W$ function or the error function ...

And then there are things that are just personal preferences for different notation (or cultural preerences - I personally have great difficulties reading something as simple as a long division if it is written the "American way" that looks to me rather like a $\sqrt .$)