That doesn't make sense though, the part about being the heir to the Septims, that means at some point the Emperor in Oblivion was doing the nasty with the other races since the Dragonborn can be Argonian if someone picks that race. I don't know it is weird thinking about that, how are you the Dragonborn if Martin Septim died without children, that means his dad had to have had other illegitimate kids that went of having kids until it gets to you, it makes no sense.
Well, Martin
did have some wild times as a worshipper of Sanguine at one point, so it's within the realm of possibility (however slim) that he had at least one illegitimate child roaming around he wasn't aware of. Likewise, if you consider the possibility that many of the Septims were fond of...erm, sowing their wild oats shall we say, despite Oblivion's claim of "zomg there's no Septims left except Martin" (which is an issue for another thread...we won't go into my issues with Oblivion's MQ

), there's probably still some descendants of the line roaming Tamriel.
Although I suspect P!nk's "heir to the Septim dynasty" comment wasn't meant to be taken literally. It's just that the Septims were Dragonborn, and the hero of Skyrim is Dragonborn, so there's a spiritual kinship if not an actual genetic one.
Though even then, I'm not sure just being Dragonborn automatically legitimizes any claim you'd have to the Ruby Throne. It's been 200 years since the last Dragonborn Emperor and the idea's probably largely fallen out of favor, or at least popular/cultural memory...particularly in predominantly human provinces like Cyrodiil or Hammerfell, where there hasn't been a Septim on the throne in living memory. While us players still have strong memories of the Septim Empire, the folks living in Tamriel in 4E 201...not so much.
Besides which, having a Dragonborn/Septim Emperor is no longer necessary, since the Dragonfires no longer need to be lit to keep those pesky Daedra at bay. Martin's sacrifice and the big honking statue in the Imperial City took care of that. So the "but you need a Dragonborn ruler" argument goes straight out the window.