First, backgrounds are nothing like a feat.
Second, how do you justify suddenly having gained social status via taking a feat (noble)? Or suddenly having the local peasants know and love you (folk hero)? Moreover, what exactly does it mean if you have both the Noble and Beggar backgrounds? That doesn't make sense. (I know, there's no Beggar in the PH, but let's take it for granted that there are tons of custom backgrounds already out there, as well as tons more waiting for some gamer to quickly write up.)
Beyond that, if you compare it to the Skilled feat, a background is significantly better. You get two skills, two languages or tools and a feature. No, I wouldn't allow it. The training rules provide for learning new proficiencies in tools and the like, and the DMG lays out that some groups will allow it for learning new skill proficiencies too. That's the route I'd take.