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The 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.

(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 $\partial D^2(r)$.)

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].

(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 $\partial D^2(r)$.)

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.

(As a particular case, which however is not so 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 $\partial D^2(r)$.)

The 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.

(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 $\partial D^2(r)$.)

The 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.

My first claim is that we can cover the domain $U$ by closed analytic discs contained entirely in $U$. Here it is crucial that the boundary is smooth, in order to produce analytic discs which are tangent to the boundary at every point. (In order to construct such discs, one can e.g. use a smooth approximation of a closed curve by analytic such curves.) Since $\phi$ restricted to each such disc has an extension by the above procedure, standard analytic continuation thus produces the sought extension of $\phi$.

(As a particular case, which however is not so 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 $\partial D^2(r)$.)

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.

(As a particular case, which however is not so 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 $\partial D^2(r)$.)