I know humor is subjective, but… really? There are people who don’t get that Biden has a big streak of old white guy combover embarrassing crazy uncle blowhard buffoonery? The guy’s given off that vibe his entire career, and it’s only gotten stronger in recent years.
As for his tendency to put his foot in his mouth, that goes way back too, long before the 2008 campaign. It’s just that fewer people outside of political geek circles noticed it until they finally paid attention with his VP nomination.
Not making a partisan point-- one could surely get away with making a Republican the subject of these jokes (no one fitting this particular set of stereotypes comes to mind right now… George W. Bush would have worked but he was quickly pegged as earnestly dumb in popular culture versus inappropriately jocular). With Biden, it’s a variation on the popular “Billy Clinton” jokes, mixed with the random non-sequiturs the Onion specializes in.