A partial answer is yes.

Given that the domain has a real analytic boundary the construction is standard. A Schwarz reflection can be performed locally near each boundary point, which together with normal analytic continuation produces the sought extension.

In general, I propose the following approach. Assume that $f(D^2) \subset g(D^2)$. The critical points of $g$ (and $f$) form a discrete set by the aforementioned similarity principle. Away from the critical points $\phi^{-1}(\mathrm{Crit}(g))$, we can write $\phi=g^{-1} \circ f$ and in this way obtain a unique continuation along small punctured discs covering the boundary of $U$. However, since $\phi$ is bounded along the boundary by assumption, the removal of singularities theorem can be applied to complete $\phi$ over the punctures.

**Remark.** At least in real dimension four, the assumption $f(D^2) \subset g(D^2)$ is not too severe since two unparametrised pseudo-holomorphic curves intersect in a *discrete* set, as was show in [[McDuff; The Local Behaviour of J-holomorphic Curves in Almost Complex 4-manifolds]][1].

(As a particular case, which however is no longer relevant given the new general formulation of the question: Every biholomorphism $\phi$ of $D^2(r)^o \subset \mathbb{C}$ which has a continuous extension to $D^2(r)$ extends to a *biholomorphism* of the *Riemann sphere*. Here we use a Schwarz reflection along $\partial D^2(r)$.)


  [1]: https://www.google.se/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CCAQFjAAahUKEwjIqe_S1d3HAhXiD3IKHYaEAIk&url=http%3A%2F%2Fprojecteuclid.org%2Feuclid.jdg%2F1214446994&usg=AFQjCNFt-m8b1Y8S5ByNlkcYGliYj-cMIQ&sig2=zsERC9XMhA0bzk1PV_IySQ&bvm=bv.101800829,d.bGQ