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consider two oblique oblique basis vectors of unit length $vecr_1, vecr_2$

then any vector $vecv = pvecr_1+qvecr_2$

define the dot product between two vectors a and b as $|b|$ (ie length of b) * (the length projection of(say) a on b made by an oblique line from a on b with angle same as that formed between $vecr_1, vecr_2$)

ie such that $vecr_1 cdot vecr_2 = 0$

this is in contrast to the project of a on b along a right angle as done with orthogonal basis where you can use the cosine.

It's easy to show that this definition of the dot product is indeed distributive.

from our definition of v

$vecv cdot vecv = p^2 + q^2 $

Then $|v|^2 = p^2 + q^2$

which geometrically isn't the case,because p and q are oblique components, not perpendicular ones, this should not give you the length of v.

Let me expand on what Yves Daust is saying with a specific example.

Let $vec r_1 = (1,0)$ and $vec r_2 =left(frac sqrt 22, frac sqrt 22right)$ and suppose we consider the vector $vec v = vec r_1 + vec r_2$.

By the definition of your inner product, $|vec v|^2 = 1^2 + 1^2 = 2$.

Now if we express $vec v$ in terms of canonical coordinates, it is $$vec v = vec r_1 + vec r_2 = (1, 0) + left(frac sqrt 22, frac sqrt 22right) = left(1+frac sqrt 22, frac sqrt 22right)$$

And therefore its Euclidean norm is given by $$|vec v|^2 = left(1+frac sqrt 22right)^2 + left(fracsqrt 22right)^2 = left(1 + sqrt 2 + frac 12right) + frac12 = 2 + sqrt 2$$

So very clearly, $|vec v| ne |vec v|$.

$|vec v|$ is a norm. It satisfies all the properties necessary to be a norm. But the key word here is "a". There are many, many, norms that can be placed on $Bbb R^2$, so simply being a norm does not make it equal to that other more famous norm.