spectroscopy - Why is $g$ tied to the oscillator strength $f$ in $log{gf}_{odot}$

You don't make it clear, but you may be confused about what $g$ is. It is the statistical weight of an atomic energy level.

The $gf$ value, which refers to a particular transition between two energy levels in an atom/ion, is used because there is symmetry in terms of emission/absorption processes once the statistical weight is taken into account.
$$g_1 f_{12} = - g_2 f_{21}$$

If you just quoted $f$, then you would also need to know what the appropriate statistical weight was in order to calculate transition probabilities.