= Solution
First recover the induced boundary metric from the <boundary distance function>. If $c(t)$ is a smooth boundary curve with $c(0)=x$ and $\dot c(0)=w\in T_x\partial M$, the local metric expansion of distance gives
$$
\lim_{t\to0}\frac{d_{g_i}(x,c(t))^2}{t^2}=g_i(w,w).
$$
One can obtain this expansion in local coordinates from the fact that the metric tends uniformly to its value at $x$; distances to nearby points have that constant metric's first-order norm. Equality of the two <boundary distance functions> makes the two quadratic values equal for every boundary-tangent $w$. Polarization therefore recovers the full bilinear induced metric:
$$
h=g_1|_{T\partial M}=g_2|_{T\partial M}.
$$
This does not yet identify the metrics on normal vectors or mixed tangent-normal pairs.
Let $\nu_i$ be the inward unit normal for $g_i$. The <boundary normal coordinates>
$$
c_i(x,r)=\exp_x^{g_i}(r\nu_i(x)),\qquad 0\le r<\varepsilon,
$$
are <collar neighborhoods> for a common sufficiently small $\varepsilon$. The <Gauss lemma> gives $c_i^*g_i=dr^2+h_i(r)$ with $h_i(0)=h$. Hence the local boundary-fixing map $c_2\circ c_1^{-1}$ already satisfies
$$
g_1=(c_2\circ c_1^{-1})^*g_2\quad\text{on }TM|_{\partial M},
$$
because both pulled-back metrics there are $dr^2+h$.
To justify a global <diffeomorphism>, rather than assume a local collar map extends, interpolate the positive metrics by $g_s=(1-s)g_1+sg_2$, $0\le s\le1$. Their inward unit normals and exponential maps give smoothly varying collars
$$
C_s(x,r)=\exp_x^{g_s}(r\nu_s(x)).
$$
Compactness of $[0,1]\times\partial M$ gives a uniform collar size after shrinking $\varepsilon$. On the image of each collar define the time-dependent <vector field>
$$
Z_s(C_s(x,r))=\partial_s C_s(x,r).
$$
Multiply it by a smooth cutoff which is one for small $r$ and zero before the collar's outer edge, and extend it by zero to the rest of $M$. This gives a smooth global time-dependent <vector field>. It vanishes on the boundary, because $C_s(x,0)=x$ for all $s$. Its flow $\Psi_s$ exists for the full interval and fixes the boundary. For sufficiently small fixed $r$, the curve $s\mapsto C_s(x,r)$ solves this flow equation, so uniqueness gives
$$
\Psi_s(c_1(x,r))=C_s(x,r).
$$
Here $C_0=c_1$ and $C_1=c_2$. Thus the final map $\psi=\Psi_1$ agrees with the local map from the $g_1$ collar to the $g_2$ collar near the boundary. In particular $D\psi$ fixes tangential vectors and sends $\nu_1$ to $\nu_2$ there. Since the tangent-normal splittings are orthogonal and the normals have unit length,
$$
\boxed{\psi|_{\partial M}=\operatorname{id},\qquad g_1=\psi^*g_2\text{ on }TM|_{\partial M}.}
$$
The construction only needs equality of the induced boundary metrics; the equal-distance hypothesis supplies that equality.
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