I am aware of the following:
Let $ X $ and $ Y $ be projective plane curves, with $ X $ nonsingular and not contained in $ Y. $ Then the sum of the multiplicities of intersection of $ X $ and $ Y $ at all points of $ X \cap Y $ equals the product of the degrees of $ X $ and $ Y. $
This is a form of Bézout's theorem, but I don't know how relevant this is to the question since the question concerns the maximum intersection multiplicity of two nonsingular conics at a particular point of intersection.