ARE_THE_SAME statements tell ASCEND that two variables should be merged such that they actually refer to the same thing: when one changes, the other changes at exactly the same time. One can still refer to the variable by either name, but it will only occupy a single place in the solver's list of variables.
For examples see Object-oriented modelling.
Note that there are some cases, such as within parameterised models, where ARE_THE_SAME is not allowed. In these cases, you can often achieve what is required using ALIASES instead.