There are probably too many such $(M,g,I_+,I_-)$ to really expect a 'classification'.  

For instance, consider the case when a complex manifold $(M,I_+)$ has real dimension $4$, and the $I_+$-holomorphic tangent bundle $T'M$ splits as the sum of two holomorphic line subbundles $T'M = L_1\oplus L_2$.  In this case, one can define an $I_-$ by reversing $I_+$ on the line bundle $L_2$.  Now let $\Omega_i$ for $i=1,2$ be a smooth real (1,1)-form of complex rank $1$ whose kernel is $L_{3-i}$ and is such that $\Omega_i$ restricts to $L_i$ to be a positive $(1,1)$-form (with respect to $I_+$).  Let $g$ be the Hermitian metric (with respect to $I_+$) that is associated to the positive $(1,1)$-form $\Omega_+ = \Omega_1 + \Omega_2$ (with respect to $I_+$).  Then the $2$-form $\Omega_- = \Omega_1 - \Omega_2$ is a positive $(1,1)$-form with respect to $I_-$ and it belongs to the same associated metric $g$. 

It is easy to check, using local coordinates, that the two associated Lee forms are equal:  Fix a point $p\in M$ and choose an $I_+$-holomorphic coordinate chart $(z^1,z^2):U\to\mathbb{C}^2$ on a $p$-neighborhood $U\subset M$ such that $\mathrm{d}z^i=0$ defines the subbundle $L_{3-i}$. Note that $\bigl(z^1,\overline{z^2}\bigr):U\to\mathbb{C}^2$ is an $I_-$-holomorphic chart on $U$.   There exist functions $u_i$ on $U$ such that
$$
\Omega_{\pm} = \tfrac{\imath}2\,\mathrm{e}^{u_1}\,\mathrm{d}z^1\wedge\mathrm{d}\overline{z^1}\pm \tfrac{\imath}2\,\mathrm{e}^{u_2}\,\mathrm{d}z^2\wedge\mathrm{d}\overline{z^2},
$$
and their common Hermitian metric is $g = \mathrm{e}^{u_1}\,\mathrm{d}z^1\circ\mathrm{d}\overline{z^1}+ \mathrm{e}^{u_2}\,\mathrm{d}z^2\circ\mathrm{d}\overline{z^2}$.
Then $\mathrm{d}\Omega_\pm = \theta\wedge\Omega_\pm$ where
$$
\theta = \frac{\partial u_2}{\partial z^1}\,\mathrm{d}z^1
        +\frac{\partial u_2}{\partial \overline{z^1}}\,\mathrm{d}\overline{z^1}
 +\frac{\partial u_1}{\partial z^2}\,\mathrm{d}z^2
        +\frac{\partial u_1}{\partial \overline{z^2}}\,\mathrm{d}\overline{z^2}.
$$
Thus, $\theta_+=\theta_-=\theta$, as claimed.  

Note that this local normal form is unique up to replacement of $z^i$ by another holomorphic function of $z^i$ for $i=1,2$, while the choice of the two functions $u_i$ is arbitrary.  It's not hard to show that, in dimension $4$, this Ansatz the only way that examples with equal Lee forms can arise.  (É. Cartan would say that the local examples of bi-Hermitian metrics with equal Lee forms in dimension $4$ 'up to diffeomorphism' depend on two arbitrary functions of four variables.)

In higher dimensions, I expect that the analysis is more complicated, but, again, there will be too many examples to really classify because having equal Lee forms is a very underdetermined set of PDE on the data $(M,g,I_+,I_-)$.