I think this is a good question because each race should have different view, thought, feeling and value on things, so based on your character race, what being a Dovahkiin means to you and your race? Sure it also depends on individual, doesn't mind if it is individual or not.
My official Dovahkiin is a Breton. For me, it is honorable title for my people, being a Dovahkiin it give my people honor in Skyrim that is the land of the Nords, I also protect High Rock interests by being a Dovahkiin in Skyrim. For Nords, Dovahkiin is something sentimental as it's tradition is back to Talos/Tiber Septim. It also mean something to the Imperial, because for a long time Imperial emperors are Dovahkiin, but now a Breton is one. In either way, as a Dovahkiin, both Imperial and Nords respected me (even though the game never show that, oh Bethesda....)
In Season Unending, sit at the table at the end table side with Imperial, Stormcloak, Greybeards, Thalmor and Blades is something you know, as a Breton. All respected me and asking my opinion on things and what i say give weight as a Dovahkiin. I can shape Skyrim landscape from that meeting. You know...a Breton...decide things on Skyrim and my decision count. Even the Thalmor can't say anything.
So what it means to be a Dovahkiin to your character race?