In this answer, formal intersection appears in this context:
Let each intersection $U_{i_1} \cap \cdots \cap U_{i_n}$ be denoted as $U_{i_1 \cdots i_n}$. Treat these sets as symbols (i.e., formal intersection), and distinguish $U_{ij}$ from $U_{ji}$ even though they are the same set.
How can the notion of "formal intersection" be made precise? I found nowhere a definition of it. I mean a similar definition as that of formal sum.
Edit: I mean a definition using only well-defined mathematical objects, such as set, function, etc, and no reference to the notation. Just like in the case of the definition of formal sum in the referenced Wikipedia article.
When something is said to be a "formal sum" or "formal intersection" or "formal" anything, it is the same thing. It just means, treat it as a symbols only instead of as having the normal meaning.
So you are familiar with the formal sum $$ 2 + 3, $$ which means the sequence of symbols $2, +, 3$ (with "$+$" in particular having no defined meaning, and being just a symbol.) Similarly, here we have $$ U_{ij} $$ with $U$ having no defined meaning, and being just a symbol with two subscripts. (In particular, the order of the subscripts is significant.)