Geometric meaning of different dot products So, the standard dot product is defined by the $I$ matrix. If using this dot product between two vectors gives me the result $0$ i can say the vectors are perpendicular because $0$ is the relative lenght of the projection of the first vector on the second one.

If i have a different dot product defined by another matrix, what is the change in his geometric menaning?
You speak of length and perpendicularity (=angle), but those concepts are defined only after a scalar product is given. Geometrically, assigning a scalar product on a vector space amounts to specifying which bases are orthonormal. Once one has an orthonormal base, one can measure lengths and angles. 

The "standard dot product" on $\mathbb R^3$ prescribes that the "standard base" 
$$
(1,0,0), (0,1,0), (0,0,1)$$ 
is orthonormal. But that is only a choice. If you choose a different scalar product, that base is not going to be orthonormal.
Suppose we have drawn the two vectors so that their tails are at the same point. The angle between the two vectors has been labelled $\theta$\

$a.b=|a||b|\cos(\theta)$ 

$|a| , |b| $ is the modulus

$\theta$ is the angle between $a$ and $b.$

we sometimes refer to the scalar
product as the dot product.

If  the scalar product of
two vectors which are at right-angles is always 0.

We say that such vectors are perpendicular or orthogonal