Right off the bat: Yes, droids are capable of telling creative narratives. C-3PO does so in Return Of The Jedi when recounting the group's past adventures to the Ewoks. He's not just reporting facts; he speaks with the emotion of a narrator, and provides sound effects for the lightsabers and carbon freezing (in front of Han Solo, proving that droids can also be assholes). But can Artoo do that? Why not? In A New Hope, he carries information, and tricks Luke into removing his restraining bolt so he can sneak off to deliver Leia's message to Obi-Wan Kenobi. Artoo can remember things, act on his own free will, and be a clever and sassy little bastard while doing it.
And this is exactly what he was designed to do. According to Wookipedia, R2-D2's line of astromech droids was designed to think creatively in order to solve problems. He could easily use that creativity to put a spin on his account of past events. Like the parts involving Artoo himself.
So Artoo worked with Luke's dad, before coincidentally being sold back to Luke years later? Far-fetched, but it's a small galaxy. But wait, rocket-boosters? There is no hint of any droid having such a feature in either trilogy until Artoo randomly whips them out during an action sequence in Attack Of The Clones, like a character in a bad fanfic.
When it comes to Mary Sue red flags, those rockets are just the start of it. Artoo isn't just present for all these events, but constantly playing a crucial role and receiving praise from important people. First, while serving Queen Amidala, he's the only droid to miraculously survive an attack, and saves her ship, earning the queen's personal thanks. He then goes from being the queen's favorite droid to the squire of the most important Jedi Knight of the era, Anakin Skywalker. And then he somehow manages to become the vessel through which Leia relays her desperate plea to Obi-Wan. And then he befriends and tags along with Leia's brother, a another powerful Jedi and the hero of the story.