A while back I was playing a new Skyrim character and I had just killed the first dragon. As I returned to whiterun I heard 'dovahkiin' being shouted. But I heard it in three syllables- dov, ah and kiin. 

The dragon born ends up seeking out the man who called his name; dragons do that when their name is called. Furthermore the name makes sense 'ah' being hunter and 'dov' being wyrm. It may be a man-made shout but so was dragon rend. 

If there is some major gap in my logic can somebody please fill it?

-Robyn

I think it's more along the lines of how you summon Odahviing and Durnehviir. Shouting a dragon's name seems to be how dragons call for each other. I'm pretty sure that's how you get Odahviing to Dragonsreach in the first place, because calling a dragon's name is like a challenge (may need confirmation though, I read a lot of fanfiction).

Also, Dovahkiin means "Dragon born", with Dovah being one word meaning a single dragon.

So I think it's not a shout in the same sense as Yol Toor Shul, but more like how people's names are considered words even if they don't mean anything.

Other than Arngeir, the Greybeards' voices are too powerful for speaking.  If they speak, the mountain trembles.

They do "summon" the dragonborn, but it is more their way of sending a letter than a Shout.

No, I don't believe it is a shout. Shouts are kind of like, spells of sorts. When the greybeards call teh protaginists name, it has no magical properties, outside of being really loud I guess.

No. No it is not.

Ruvgein

Other than Arngeir, the Greybeards' voices are too powerful for speaking.  If they speak, the mountain trembles.

They do "summon" the dragonborn, but it is more their way of sending a letter than a Shout.

if one wants to be technical...dovahkiin can be broken down into a shout, "dov ah kiin". both of you could be right in that regard. if a dragon wanted to challenge the dragonborn they could also do the same thing  the grebeards did. out of curiosity you will go investigate because deep down you are being called to that location. maybe that curiosity stems from the dragon soul or maybe its just you seeking out the originator of the one who called for you.

well maybe the laws are diferent in skyrim.

Technically, dovahkiin COULD be a shout. Instead of dovah kiin, "dragon born", it could be dov ah kiin, "born (to) hunt dragonkind". It would still be a title, but aren't dragon names just descriptions of what the dragon has accomplished, their personality, or their fate (such as Alduin's name)?

So maybe the Greybeards calling the Dovahkiin is them using "dovahkiin" as a shout rather than just "dragon born"..