Is there a name/word for his type of voice?

It sounds like his voice is breaking, like he his is straddling the line between normal chest voice and falsetto.

Aaron Neville has been described by critics as possessing one of the most splendid falsetto-tinged tenors in pop music. The singer, who is deeply religious, puts it another way: “I describe it as God in me touching God in the person that hears it; at least that’s what it feels like to me.” – Sheila Rule

What is even more distinctive is the complexity of the delicate, half-yodeled melismas that Mr. Neville interpolates into many of his songs. – Stephen Holden
Source

It sounds kind of like yodeling. I have heard some other country singers do it too. “Falsetto tinged” is a great way to describe it but I think @chocolatechip has well-described what is happening. Neville does have great control when using it. Randy Travis has a similar ability.

Sorry, Neville fans, but I’ve grown weary of his voice. I once enjoyed it—like on “Don’t Know Too Much” with Linda Ronstadt—but now I find it grating in the same way I do so many female singers these days who scream lyrics and unnecessarily manipulate them. You hear them endlessly in supermarkets. I dunno, maybe my years are getting old, but I prefer vocalists who sing softer and intimately. S

@Austinlad Didn’t say I could maintain a diet of it. I share your sentiments described above. I am probably getting old too.

Truely a remarkable voice!
My favorite Neville song is here
Oops, I mean here

There are many musical traditions that incorporate this kind of “breaking” into the music. Middle Eastern music is one of those traditions—it can be very mournful and sound a bit like crying.

@chocolatechip I agree, it almost sounds like a very pronounced and very low vibrato.

Castrato

@gailcalled

Haha, that’s great.

@Foolaholic: Actually, that was my cat, Milo’s, suggestion.

It thnik it’s combination of a breaking voice and falsetto. I love country voices.

@Megaperceptiva You are mega perceptive. ;-) Welcome to fluther.