I am studying lambda calculus for the first time and I was trying to do the reduction beta of the lambda term $(\lambda x.xy)y$. Can I assume that these two free variables $y$ are the same? Or do I need to rename the rightmost $y$?
$y$ is free in the term you have given, so it is indeed the same variable.
You would not have the same variable in case $y$ was bound by the lambda. That would be the case, for example, if the term were to be $(\lambda y.xy)y$; then the rightmost $y$ would be free, but the one in $xy$ would be bound by lambda and, hence, a different variable.