Huge character breakdown incoming so ignore if you want to, although I would appreciate the read. Also, huge warning, there are OPINIONS ABOUT DRAGON BALL in this so procede with caution lol.
No, they aren't the same character, infact, they are very different in how they are written. I'll start with Vegeta, then Bakugo, then explain why they are soooooo different.
Vegeta is the prince of all Saiyan's, his pride is driven by the heritage of his people and the tragedy of his races extinction. His aggresiveness and drive to get better is driven by the two afformentioned reasons but also the guilt he has about not being able to stop it. He is built up as a super strong character for his world, and continues to grow alongside Goku. Vegeta's story is all about having pride in your heritage and realising that you aren't better than everyone, a story of how to balance pride and tolerance.
Bakugo is the golden child, the prodegy and what people see as the future number 1 hero. His pride is because of built up compliments and praise he had aquired throughout his life. His aggressive attitude is a result of him seeing everyone around him as inferior, and someone who needs protecting due to being too weak themselves. He also grows in power alongside Deku. Bakugos story is about picking up the shattered pieces of his unending belief that Deku was weak, and learning to live with it and acknowledge his own weaknesses and short comings. His is a story of how to develop a sense of humility.
Where these two differ is how they are fundimentally written. Vegeta is built up to be so powerful to show how powerful Goku is, he is written to be perpetually weaker than Goku to emphasise how powerful Goku is. Bakugo is built up to be so powerful to show how WEAK Deku is by comparison. He is something for Deku to strive to be, a driving force for our protagonist and a very powerful motivator (even if Bakugo doesn't want to be.) Vegeta has been and always will be weaker than Goku because that's how his character is written, Bakugo will be better than Deku for a long time as he has always been to show how much Deku needs to grow. Most people see these two and think "well they yell and are angry and have spiky hair so they MUST be the same" but I hope I have illustrated why they aren't. I believe that Bakugo is a better character than Vegeta, by far for one simple reason, we don't know how the rivalry between him and Deku will go. In the actual opening of the show it states that Deku will be the greatest Hero, not the number 1 hero, so Bakugo might actually end stronger than Deku as he always has. On the flip side, Vegeta will never be stronger than Goku for an extended period of time.
I'm not debating who is stronger, that isn't a debate. Vegeta is infinitely stronger than Bakugo and probably will be forever (unless adult Bakugo gets OFA or something and even then Vegeta will be alot stronger) but that is something I like about the my hero academia characters and something I don't like about the dragon ball characters. In MHA you can't just blow up the planet your on when your in a bind, your not destroying galaxies with every strike like you so in dragon ball (but I'm not saying Dragon Ball is a bad show, I'm just not into it. Like JoJo, I'm not into the show but I can tell it is a good show.) But I digress, the point is, please don't put in the comments something like "well Vegeta would still win in a fight" because I know that, everyone knows that, and if they don't then they need a new brain cuz that one is clearly failing them.
Who do you guys think is the better character, who do you think is cooler, do you like dragon ball or my hero academia more, do you think that Bakugo should be shipped with Kirishima or Mina (idk, I couldn't come up with another question lol but I'm choosing Mina, idk why.) A long read I know, but thank you for reading.