If we start from the definition of the notion of families of varieties ( or algebraic ( projective ) varieties more precisely ), as proposed by Joe Harris in his book: Algebraic geometry: A first course, page: $41$ : Is the concept of vector bundle, ( i.e: locally free sheaf ), or projective vector bundle is a special case of the notion of family of varieties ( or algebraic ( projective ) varieties ) ?
If so, why then the concept of tensor product between two vector bundles is replaced by the notion of fiber product over a base in the case of : families of varieties ( or algebraic varieties more precisely ) ? In other words, does a the notion of fiber product over a base coincide locally with the notion of tensor product ? How can we express this mathematically ?
Thanks in advance for your help.