What do two non-terminal symbols on the left hand side of a BNF grammar mean?
That seems unorthodox to me, but I'm certainly not current with all the formal language theory textbooks being used these days.

The point is that in 

$$\langle expression\rangle :=\langle number\rangle \;| \;\langle expression\rangle\langle operation\rangle\langle expression \rangle$$

the two instances of $\langle expression\rangle$ on the right-hand side are unrelated to each other. In a derivation, they are derived independently and nothing constrains their derivations. Sometimes we need to refer to them (for example, when we get to semantic analysis) and then it might be convenient to give them different spellings.

I would normally use subscripts:

$$\langle expression\rangle :=\langle number\rangle \;| \;\langle expression_1\rangle\langle operation\rangle\langle expression_2  \rangle$$

but your professor seems to want to be able to say that $e$ and $f$ are aliases for the same non-terminal (but not the same derivation). This seems confusing to me. Is it really so hard to write:

$$e \Rightarrow n \; | \; e_1\; o \; e_2$$