Traditionally, multiplication takes two elements of a set and turns them into another element of that set. The dot product doesn't do that, it takes two vectors and turns them into a scalar.

The cross product only exists in three dimensions. And it's not associative (A×B×C gives a different answer depending which order you do it in), which is another thing multiplication usually satisfies.

There are two other not-quite-multiplication operators that I recall seeing. There's an analog of the cross product in two dimensions: (a,b,0)×(c,d,0) = (0,0,ad-bc), so it can be useful to have an operator (a,b)×(c,d) = ad-bc, again turning two vectors into a scalar.

And if the dot product is defined in terms of matrix multiplication by A·B = AᵀB, then you can also define an operator ABᵀ, turning two vectors into a matrix. These vectors don't even need to have the same length.