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So, there were more than a few people that wanted an explanation of how things work with the Blood Adoption and Harry and the genealogy parchment in Book One. This is the best that I can explain it and I hope that it helps.
James Blood Adopted Harry. A Blood Adoption does not take away from a child... it only adds to it. But, because he is not a full blooded Potter... he could get Sanguinem maledicta. That is how he could be the Potter Heir... Just like how he can be the Black Heir even though he wasn't born a Black. So... Harry has a bunch of parents
Blood Father: Severus Snape
Blood Mother: Lily Potter nee Evans
1st Blood Adoptive Father: James Potter
2nd Blood Adoptive Father: Sirius Black
Secondary Adoptive Father: Remus Black nee Lupin
So, He is Heir Black (the strongest of the 3), Heir Potter, Heir Prince and the eventual Lord Black, Heir Potter, Lord Prince (when he reaches twenty-five, Sev will step down as Lord Prince)
Draco will hold the title of Consort Black, Lord Malfoy eventually but nothing else. The Potter and Prince Consort titles don't really matter to them. He would be Consort Potter and Prince, but not really as the titles of Lord Potter and Lord Prince are being held for Harry's children. The titles are really being held until the children are born and reach the age of eleven for the Potter Line, or in the case of the Prince Line... seventeen. Then... the first born becomes Heir Black (if a male), the second becomes Heir Malfoy~if a boy~ (AND if Draco is Lord and Luce is dead or steps down), the third will be Heir Potter until eleven then Lord Potter (or Lady if a girl), and the fourth will be Heir Prince until seventeen then Lord or Lady Prince (Harry will be in his 40's or 50's by then, he won't want to have the dual responsibility.
After Harry and Draco's children get married, things will revert to normal Pureblood traditions (Heir until parent dies, abdicates, or goes to Azkaban).
Now, as far as Blood Adoptions go, there are two types. One that was used by James (in this story, and Sirius in DCH) and the one that was brewed by Severus for Siri.
The Blood Adoption potion that James Potter used the day that Harry was born was one that allowed for a child to become Heir of a line they were not born into. It did not add much to Harry, and other than an Heir Parchment Test, wouldn’t show up (and the only ones legally allowed to perform that test? You guessed it… The Gringotts Goblins). It did not allow Harry any of James’ traits or features. He was still Biologically for all intents and purposes, Severus and Lily’s son. He held the Potter name, because Lily was married to James at the time of Harry’s birth. He would still have held entitlement to Lord Prince at Seventeen (if Sev wanted to give it to him). He may have been told once he reached Hogwarts age, but James may not have wanted to tell him. We will never know. They really may have waited until Harry reached the age of majority in the Wizarding World (Seventeen) and Harry went for Heir testing with the Goblins (a requirement of anyone in an Ancient or Noble House at that age). Harry, I think would have been upset in finding out that way, but I am sure that Severus would have been on the fringes of his life, so it may not have hurt too terrible. Again, all of this is speculation and conjecture since James died so early.
The Blood Adoption potion that Severus brewed for Siri was something he created especially for this instance. It was meant to mask the fact that Severus was Harry’s Father. It didn’t take away all of Severus’ blood from Harry, but enough that it would move him down to the levels that James’ blood was in Harry. This is why the potion affected Harry like it did. It changed him from mostly Neutral Magic (Harry was never a Light Wizard… even in Canon~ I don’t care what JKR says~), to Dark Magic. It gave him features of a Black. It may not have erased Lily’s contributions, but if you put Harry Potter and Harry Black in a room together… They would look like distant cousins. Which, technically they would have been, since Harry Potter’s grandmum was a Black.
Eventually Remus will need to be factored in here, but as explained, he cannot Blood Adopt Harry. So for now, he is a parent like an Muggleborn child would have an Adoptive Parent. He is legally Remus’ just not Blood.