I am not sure if this is an inversion or some other chord. Notation is Cm6. Is this an inversion of C minor chord? Or is it a sixth chord of some sort - C minor with a sixth added? Key signature in the song is G major.
Cm6 is C Eb G A. C minor triad with an added sixth. It's one of the extended chords that doesn't fit into the "pile of thirds" hierarchy, it's not a Cm13 (C Eb G Bb D [F] A) with missing notes!
And it's just fine in the key of G. Here's a common, corny even, useage - firmly within a G major tonality.
Yes, according to the most common use of such symbols, in popular music and jazz transcriptions, Cm6 would be C minor chord with an added 6th (i.e. a natural A).
The figure 6 (written as superscript) can be used to mean a first inversion of a major or minor chord, but that is usually in the context of the so called "roman numeral analisys" of tonal harmony. So in that context a roman numeral in relation to the tonality, not the note name, would normally be used to designate the chord.