Does common eigenvectors between two matrices A,B implies some property for the vectors? If there are two matrixes that they have common eigenvectors for some eigenvalues that implies that those two matrixes are identical? What can we say for those two matrices?
 A: If the two matrices have the same eigenvalues with the same multiplicity they have the same characteristic polynomial. If the multiplicity of these eigenvalues equal the dimension of the eigenspaces (vector sub-spaces) then the two matrices are similar to a diagonal matrix up to a change of base. This similarity is a transitive property and then you can say that the two matrices you have represent the same endomorphism but in two different bases of the same vector space. The underlying endomorphism is the same but the matrices are definitely NOT identical. Remember that a matrix is a way to represent an endomorphism after one has chosen a  vector base.
Edit:
Similar matrices $A$ and $B$ are such that there exists an invertible matrix $P$ such that: $A = PBP^{-1}$
A: If two matrices have the same set of eigenvectors but different eigenvalues, then they can be simultaneously diagonalized, which means that the two matrices commute which each other, that is if the two matrices are A and B, AB = BA.
A: If the two matrices are diagonalizable, then they must be equal. 
Explanation:
Let $P$ be the matrix whose columns are a basis of eigenvectors of $A,B$. Then $P^{-1}AP=D_1,P^{-1}BP=D_2$ where $D_1,D_2$ are daigonal matrices with the (mutual) eigenvalues on the diagonal. Now we just need to make sure that the orders in which they are placed on the diagonal are the same. But these orders are in correspondence to the order of the columns of $P$ so we finished.
In particular this is true for any two symmetric positive-definite (s.p.d) matrices. This uniqueness result is used to prove the uniqueness of the positive square root of a s.p.d matrix.
