The thing is that proportionality means a constant in the formula. $R$ is a constant, but $P$ isn't.
Someone (Ohm) found out that $V$ is proportional to $I$: $$V \propto I$$ He found out by testing and experimenting. Everytime $V$ was doubled, $I$ was also doubled. This is what is called (direct) proportionality. We can write it with a proportionality constant, for example called $R$: $$V=RI$$ $R$ is constant, which causes the doubling of $V$ in our circuit to double $I$ as well.
Someone then also found the relationship: $$P=VI$$ by testing and experimenting. And yes, we could write this as: $$V=P\frac1I$$ which at first sight looks like inverse proportionality. Double the $V$ should then half the $I$.
But $P$ is not constant. If we change $V$ in our circuit, both $I$ and $P$ change as well. Doubling $V$ does not half the $I$.
Conclusion is that $R$ is a proportionality constant, while $P$ certainly isn't (because it isn't constant). It looks like inverse proportionality at first sight, but isn't.