What does the intersection symbol (∩) mean when applied to two non-set elements?
Eh.  $\cap$ doesn't mean anything on non-set variables.  Looking at the paper, that part of the paper doesn't make any sense to me either.  The paper says $x_1,x_2$ are Boolean variables, but the notation $x_1 \cap x_2$ is not well-defined on two Boolean variables; you can't take the set-intersection of two Boolean variables.  If $x_1,x_2$ were sets somehow (which the paper says they aren't), then the meaning of $x_1 \cap x_2 \in \{x_1,x_2,v\}$ is that either $x_1 \subseteq x_2$ or $x_2 \subseteq x_1$ or $x_1 \cap x_2 = v$.

So it's anyone's guess what that formula was supposed to represent.  If you absolutely must understand the paper, you'll probably have to read the surrounding context to guess what the intention was.  Better yet, find a better paper to read, if you can.  In any case, the reason you are confused is because what's written is confusing and seems like mathematical gibberish to me.  It happens; no one is perfect.  Don't assume that just because a paper is published in a journal, that what's written there will necessarily be correct or useful or meaningful.