It’s basically just a joke. I got it from a professor of mine. But like all good jokes, there’s some element of truth in it: most problems in applied maths relate to something physical, and in (non-quantum) physics, something happens and only one thing happens. Therefore your problems are either bad models of reality or they have exactly one solution. It often turns out that even if your methods aren’t technically correct or properly justified, you can still reach that solution.

It is reminiscent of what Euler called “the universality of analysis” (note that over the history of mathematics, analysis has been used to refer to just about everything that isn’t geometry. In this case it means roughly the sort of mathematics a physicist or engineer might know well, so non-abstract algebra with integration and differentiation). He could do crazy things like solve combinatorics problems using infinite expansions of series in ways that didn’t really make sense (I think this example is well known but I can’t think what it is) but get to the right answer. A non-Eulerian example would be Dirac delta functions which aren’t functions at all and can be tricky to formalise but also just magically work within the framework of abstract symbol manipulation that is used in applied maths without significant modification (except when e.g. some fundamental property of your model of the universe depends on the value that the Heaviside step function takes at 0)

> over the history of mathematics, analysis has been used to refer to just about everything that isn’t geometry

Actually, at the start (2500 years ago) the word “analysis” in mathematics was only about geometry.

See e.g. https://www.jstor.org/stable/2250061

> I think this example is well known but I can’t think what it is

Could be the basel problem though he probably used this trick many times in his prolific life:

https://en.wikipedia.org/wiki/Basel_problem#Euler's_approach

i wonder if it's a folk theorem kind of thing? i heard something similar from one of my physics professors back in undergrad