Let me begin with a quote from Lee's Introduction to Smooth Manifolds, where my question came from.
Proposition 15.5. Let $M$ be a smooth $n$-manifold. Any nonvanishing $n$-form $\omega$ on $M$ determines a unique orientation of $M$ for which $\omega$ is positively-oriented at each point. Conversely, if $M$ is given an orientation, then there is a smooth nonvanishing $n$-form on $M$ that is positively-oriented at each point.
I'm wondering what the uniqueness in the statement is all about. Lee did not stress it in the proof of the proposition. All he did is show that the pointwise orientation on $M$ induced by $\omega$ is actually continuous, making $M$ orientable. However, Lee did pose a remark before the proof that seems to address my question:
Remark. ... It is easy to check that if $\omega$ and $\widetilde{\omega}$ are two positively-oriented smooth $n$-forms on $M$, then $\widetilde{\omega}=f\omega$ for some positive smooth function $f$ on $M$. ...
So the uniqueness is up to a positive function really? Thank you for your attention.