I have read that adoption was treated very seriously by the ancient Romans where adoptees would be considered equal to any biological children. Was this also the case in the above situation? If not, what made this different. If it was, were Livia’s responsibilities as a mother/sister blurred at all?
For all of his Republican posturing, Augustus was determined to establish a dynasty. The testamentary adoption (posthumous adoption by will) of his wife was a means to that end.
Testamentary adoption seems to have differed in some respects from the two legal forms of direct adoption. It typically involved taking the name of one's new legal "father" (as Augustus famously had when he was adopted in Caesar's will). But it did not necessarily entail calling oneself the "son" of that man. When Pliny the Younger was posthumously adopted by his uncle Pliny the Elder, he took his uncle's name (Gaius Plinius Caecilius Secundus) - but continued to describe himself as the son of Lucius (his natural father). After her adoption, likewise, Livia was still described in at least some inscriptions as the daughter of Marcus Drusus.
The Romans, in other words, were under no illusions that testamentary adoption fundamentally changed someone's social responsibilities or identity. They did, however, recognize the real symbolic (and often economic) significance of the act - and this seems to have been Augustus' motive for adopting his wife. Augustus wanted to ensure continuity and a smooth transfer of power. By adopting Livia and making her part of the Julian clan, he could at once formalize her place in the imperial family and accentuate the legitimacy of Tiberius, now a Julian by both adoption and "birth" (since his mother was, retroactively, a Julian too).