During "Dragon Rising" in Skyrim, a Hold Guard mentioned you're 'just like Tiber Septim himself' or something, referencing the dragon blood that flows through both of their veins.

Is this saying that the Dragonborn is directly blood-related to the Emperor, Tiber Septim? In that snse, the Dragonborn is the true heir to the throne, right? OR at least, a contested one. What is this saying?

No. They're just both Dragonborn.

Of course, the Dragonborn could make a good claim for the throne, but that would make a really bad story.