Name when two functions are equal under integration (expectation)? What is it called when $E[X] = E[Y]$? That is,

$$\int x f(x)\,dx = \int y g(y)\,dy.$$

What I want to say is not that the expectation of $X$ is equal to that of $Y$ but rather (the equivalent statement) that $xf(x)$ integrates to the same value as $xg(x)$. I would like the statement to say something along the lines of:

"$f$ is [definition] with respect to $g$"

Assume that the expectations, $\int f$, and $\int g$ are finite.
I don't think equality of integrals over a single set ($\mathbb{R}$ in this case) endows the integrands with special names.  In probability theory, we would say just that $X$ and $Y$ have the same first moments, but I think you're aware of that.

OTOH since $X,Y$ are $\mathbb{R}$-valued, if you can show that $\int_a^b x\,f(x)\, dx = \int_a^b y\,g(y)\,dy$ for all open intervals $(a,b)$, then the integrands would be said to be equal almost everywhere.
I don't think equality of integrals over a single set ($\mathbb{R}$ in this case) endows the integrands with special names.  In probability theory, we would say just that $X$ and $Y$ have the same first moments, but I think you're aware of that.

OTOH since $X,Y$ are $\mathbb{R}$-valued, if you can show that $\int_a^b x\,f(x)\, dx = \int_a^b y\,g(y)\,dy$ for all open intervals $(a,b)$, then the integrands would be said to be equal almost everywhere.