I'm not certain how LE made the leap from the woman missing her common-law husband to this skeleton in fact being that man (as no DNA comparison has yet been done).
I guess it could simply be because he was carrying both his own -albeit fraudulent- ID and that of the wife. I guess Skeleton could have stolen or bought both sets but I assume LE thought this was unlikely, maybe because Skeleton appeared just as shady a character (from the way he died and was not reported missing) as the common-law husband had been.
Maybe they obtained photos from the wife?