When we have three numbers, $(a \times b)/c = a \times (b/c) = (a/c) \times b$.
The first equality looks like associativity, since we are only changing the order of adjacent operations, and the second equality (which is a consequence of the first equality and commutativity of multiplication) looks like commutativty since we are changing the order of the operations. However, Wikipedia's definitions tell us that associativity and commutativity involve only one function.
How is this property called? Does it have a name?
First, let's make something clearer: the notation $a/c$ is by definition the multiplication $ac^{-1}$ where $c^{-1}$ is the unique real number such that $cc^{-1}=c^{-1}c=1$. Now, with this in mind your identities are written $$(ab)c^{-1}=a(bc^{-1})=(ac^{-1})b.$$ The first equality is just associativity of multiplication. The second equality is a combination of associativity and commutativity: $$a(bc^{-1})=a(c^{-1}b)=(ac^{-1})b.$$