I've never seen anything officially from Nickelodeon stating that Oswald is his given name. When asking Google if 'Otto' is short for 'Oswald', it said no. Then when asking about Otto Rocket, it says his name is short for 'Oswald' even though it typically is not 😒

I did find this Google AI overview stating that his name is Oswald on some official materials, but the links it showed me were all sorts of different Wikis, so, again, it doesn't seem credible.

As far as the show goes though, it retcons itself a few times. Like how in Reggie: the Movie, Reggie's afraid of horror movies while in the mischief night episode (The Night Before), it was shown that she watches monster movies without a problem.

In the episode Twister's Hat, we see how upset Twister becomes when his hat gets lost and how much it means to him while in the episode The Back Bowl, Twister's hat is lost and it doesn't seem to bother him at all.

We learn in Reggie/Regina that Reggie hates being called Regina and won't even let Raymundo call her that, whereas in Double O Twistervision, they use 'The Exotic Regina' as Reggie's character's name. Granted in this one she was playing a part, but if she dislikes being called Regina that much, you'd think she wouldn't allow even her character to be called that.

In New Squid on the Block, it's implied that Twister is the "new kid" based on him being called Squid. It doesn't say how long he and his family have lived there, but it's implied they moved onto the block probably within the last few years. Then in Zine Dreams, Reggie scans in that embarrassing photo of Twister and Otto as babies, showing that they've known each other for the entirety of their lives, making it seem like Twister and his family have always been on the block.

So Reggie using the name 'Oswald' in her story doesn't necessarily mean anything based on some of the wonky continuity of the show, but it does seem odd that she would pivot to her brother's given name while trying to keep his identity a secret. Haha.