This is a very old dispute.

Rene Descartes introduced the idea that truth is based entirely on observation: his famous "cogito ergo sum", I think therefore I am, was to be the starting point of all knowledge: you're here because you're thinking, so start with that. The Cartesian Method was to increase knowledge through axioms derived from careful , certain observation. This is at the core of the modern scientific method. Talk with a scientist doing experimental research on a new vaccine, and you will hear at least some Descartes in what she's doing.

But a professor of rhetoric, Giambattista Vico, disputed that this was the only method of gaining knowledge . He pointed out that the Cartesian Method was good for understanding the non-human world, but not the human one. People understood things like history and the humanities rather well, because people make, construct them. A common Vico quote:

to introduce geometrical method into practical life is "like trying to go mad with the rules of reason", attempting to proceed by a straight line among the tortuosities of life, as though human affairs were not ruled by capriciousness, temerity, opportunity, and chance. Similarly, to arrange a political speech according to the precepts of geometrical method is equivalent to stripping it of any acute remarks and to uttering nothing but pedestrian lines of argument. The Ancient Wisdom of the Italians (1710)

Vico's ideas became quite important to historians.

Quite often in a discussion like this, the science nerd wants to insult the history nerd as being vague about facts, and your friend sounds like the typical science nerd in this. However, Vico also pointed out that truth is something that humans construct, not a geometrical principle. For example, it is impossible to point at two traces on an electroencephalogram (EEG) and say which one is "True" and which one is "False". Yes, you can talk about electricity; talk about measuring the resistance, current, voltage, or describe different frequencies, wave forms. But there won't be a particular squiggle for the brain thinking "false", or one for it thinking "true".

The science nerd's insult, that historians are just throwing around opinions , in this light begins to look more like whining: why can't humans be as simple as science? Why do humans have to be so complicated? At this point, perhaps the best thing is to point the science nerd over to r/philosophy/ or /r/relationship_advice/

Fosl, P. S., & Baggini, J. (2020). The Philosopher’s Toolkit: A Compendium of Philosophical Concepts and Methods

History is not a natural science, but that does not mean it is "just made up of opinions." And not all sciences rely on mathematics, either (and plenty of pseudosciences also rely on mathematics).

History is an empirical, usually qualitative, humanistic and social-scientific study of the past that expresses its conclusions in the form of narratives and formal arguments. It is not the same sort of thing as the natural sciences nor does it generally try to be. If you want to play the "does this field masquerade as a science but is really something else?" game, it is better to do so with fields that actually represent themselves as sciences, like Economics or Political Science.

One could drill down further into what a rigorous definition of a "science" (warning: the demarcation problem, as this is called, has no general solution), or what the truth-status of historical claims is (it is generally somewhere in between "objective truth" and "subjective opinion"), but that seems like a deeper discussion than your friend is ready to have about it.