Is there a specific theorem or proof in linear algebra which states the following:
For any vectors $a,\,b\,$and $c$, where $a\ne b\ne c$ and $\circ$ represents the Hadamard product:
$(a\cdot{b})\times(a\cdot{c})$ does not necessarily equal $a\cdot({b \circ c})$
Intuitively, I can prove this, but I wonder if there is a well-known, named proof already laid out.
What you need here is just a counter example.
Actually, your statement is already false in the scalar case ($1 \times 1$ vectors). Indeed, take $a = 2$, $b = 3$, and $c = 4$. Then, you have $$ (a \cdot b) \times (a \cdot c) = a^2 bc = 48 \neq 24 = abc = a\cdot (b \circ c). $$