Multiplication by One Throughout school we are taught that when something is multiplied by 1, it equals itself.
But as I am learning about higher level mathematics, I am understanding that not everything is as black and white as that (like how infinity multiplied by zero isn't as simple as it seems).
Is there anything in higher-level, more advanced, mathematics that when multiplied by one does not equal itself?

It may not be what you're looking for, but I believe this is related. In the case of limits, repeated multiplication by a limiting value of $1$ can have surprising effects.
Consider this:
$$\left(1+\frac{1}{n}\right)^{n}=\left(1+\frac{1}{n}\right)\left(1+\frac{1}{n}\right)\left(1+\frac{1}{n}\right)\ldots$$
For large values of $n$, this appears to approach:
$$\left(1+0\right)\left(1+0\right)\left(1+0\right)\ldots=1\cdot1\cdot1\ldots=1$$
$$1^{\infty}\stackrel{?}{=}1$$
However, this is not the true value of the limit. In reality:
$$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}=e\gt1$$
It appears that infinite repeated multiplication by $1$ somehow becomes another value entirely. This is why $1^\infty$ is considered an indeterminate form.
