I guess I'm unclear on what physical concept she's getting at by multiplying vectors. Depending on which concept you're trying to calculate with your multiplication, you have dot products like work and cross product like torque.
Once you put a "reasonable" set of constraints on it... you discover that you can't actually multiply vectors (no function exists that satisfies the properties you want). Though the talk isn't about proving that (or justifying the set of constraints that mean you can't multiply vectors) and instead goes off in another direction of extending your vector space to a bigger space (like how the complex numbers are a bigger space than the reals) where you can define a reasonable multiplication operator.
Another property you want (and the talk uses) is that the operator is that the operator is from V x V to something. I.e. we are multiplying two vectors (because that's what we asked for in the title) not a scalar and a vector. That excludes your counter example, but still isn't nearly enough to make it so that no multiplication operator exists.
I'll be honest and say I'm not listing properties here because I don't remember what properties are needed to make it so you can't define the operator... hopefully someone who has studied this a bit more recently or thoroughly than me can chime in.
https://mathworld.wolfram.com/Ring.html
Scalar and dot products don't stay within the group of vectors and component wise multiplication doesn't always have inverses.
I've been blanking on what exactly the interactions with scalars that we need to preserve are...
Introductory physics textbooks proceed to tell us that "well actually," the result of a cross product (such as an angular velocity vector or the Poynting vector of electromagnetism) is actually a "psuedovector."
Other formalisms treat the cross product in a more hygienic and general manner.
This is all to say that familiar mechanisms like the "dot product" and "cross product" are not necessarily as "natural" as you may have been lead to believe.
The cross product, sure: its problem is that it dualizes unnecessarily, making you deal with a normal vector when you almost always just want the plane.
But what did the dot product ever do to you?
Fair enough.
What I was getting at is that "standard vector analysis" is a choice, and it turns out that there are alternatives where things are defined differently.
I wish we would stop teaching/using the cross-product. Bivectors make a ton more sense, and as a bonus do away with the right hand rule.
1) For any vectors u,v in V, the product u*v is in V. (this rules out the dot product as a general product)
2) For u,v in V, u*v = v*u
3) For u,v,w in V, v*(u+w) = (v*u) + (v*w)
4) For u,v in V and s in F (the field V is a vectorspace over), s(u*v) = (su)*v = s*(uv)
Under these restrictions you can still cook up products, but fewer and sadder ones. In general you will not have a multiplicative inverse, for instance.
If so, may I wonder if your drop rule 2 and insist on having for most vector u there exist a v so that u*v = 1. As an icing let us say having a 0 vector. That could be something.
*It doesn't rely on rule 2, it does however rely on (u*v)*w = u*(v*w). Adding rule 2 excludes the quaternions.
v * w = v * (w + 0) # Vector space property that w + 0 = w
= v * w + v * 0 # Distributive property
Another vector space property guarantees that there exists -(v * w) such that -(v * w) + (v * w) = 0. Add it to both sides
-(v * w) + v * w = -(v * w) + v * w + v * 0
0 = 0 + v * 0 # And then reduce based on the definition of -(v * w)
0 = v * 0 # Vector space property that 0 + x = x
So you have your icing :)
(Spoiler: it's all of them)