No, you do not gain Warlock abilities.
Let's take a closer look at the text:
When you reach 3rd level, you strike a bargain with an otherworldly being of your choice: the Archfey, the Fiend, or the Great Old One, each detailed in the warlock section of the Player's Handbook, the Undying in the Sword Coast Adventurer’s Guide, or the Celestial or Hexblade in Xanathar’s Guide to Everything. Your choice augments some of your order features.
The text specifies "augmenting your order features", and then goes on to specify which features are augmented. It does not say anything about gaining other abilities beyond the ones specified in the Blood Hunter class. The choice of patron is simply used to flavor and inform some of the other class abilities.
Basically, what we have is two distinct abilities, with the same name ("Otherworldly Patron"), but different benefits. It's kind of like the Spellcasting feature - Wizards and Sorcerers both have "Spellcasting", but if we were to assume that two classes with a feature of the same name grant benefits from the other class, then Wizards would be able to cast Sorcerer spells, and vice versa.