Vectors do not have a basis. Avector spacehas a basis. A vector is represented in a vector space as a linear combination of basis elements uniquely. So if you are asking if two vectors in a vector space share the same coefficients under the same basis, then yes, they are the same.

Alternatively, perhaps you were asking if two vector spaces have the same basis, are they the same? This too is true if the vector spaces are over the same field, because they would then span the same set.