Is it possible to reduce the number of variables in bin packing?
An integer program, or more properly an integer linear program, consists of a linear program together with integrality constraints stating that some of the variables are integers. As such, its objective function is always a linear combination of variables.

When the objective is minimization, it is admissible to have $\max$ operators (appearing positively) in the objective function. This means that there is an equivalent proper integer program. This program is obtained by introducing auxiliary variables, just as the variables $y_i$ are introduced in your example to implement $\max_j x_{ij}$.

To answer your question, it all depends on what you consider as an integer program. The standard definition only allows linear objective functions, and in this case the $y_i$ are necessary. If you also allow $\max$ operators in the objective function, then the $y_i$ are not necessary.