I was wondering if any of you people with Dragonborn could confirm that the Dragon Aspect shout actually powered up the other shouts. I saw one or two playthroughs and didn't really notice any difference in power with or without the shout.

It doesn't strengthen shouts at all. It boosts armor rating, melee damage, I think mana efficiency, summons a spectral dragonborn at low health, & lowers shout cooldowns.
The shouts description doesn't say anything about powering shouts up, & I didn't notice an increase of any kind.

I think you might be confused with a perk granted by the black books and Dragon Aspect. I don't have all 3 words of Aspect yet, so I can't say for sure, but there was a perk from the books that powered up Unrelenting Force, Fire Breath, and Frost Breath.