I don't think there's a lot of distortion in that sound. This kind of funkiness is mostly achieved with a good strong technique (deep apoyando, rather close to the bridge), a low-ish action so you get a decent amount of fingerboard in the sound (but not slap – that uses a single collision, whereas this fingerstyle needs more of a uniform buzzing; in fact you can achieve this effect quite well with a fretless), and plenty of midrange in the amp dials.
Of course, if the amp is a reasonably turned up tube amp then it'll also add some overdrive, but that's not required for the characteristic feel. In fact, in my experience hybrid or pure transistor amps can be better at giving the required, very direct response. Small speakers in the cabinet also help (i.e. multiple 10" instead of one 15").
(There definitely is some overdrive, at least at the end of that Funkadelic song though.)