They are not perfectly positively correlated: Even when random variables are deterministically related (which would require $X$ to be deterministic in this case), perfect correlation requires them to be related via an affine transformation. This would require a relationship of the form:
$$V(t) = \ln \Big[ 1+\frac{U(t)}{X(t)} \Big] = a + b U(t),$$
where $a \in \mathbb{R}$ and $b>0$. Solving for the process $X$ gives:
$$X(t) = \frac{U(t)}{\exp(a + b U(t))-1}.$$
This is inconsistent with your specification that $X$ is a geometric Brownian motion.