What is different about him is that he doesn't seem to have intellectual curiosity.
The high level of competitiveness can be a red flag for this: being externally motivated. The goal, over time, is for individuals to develop internal locus of control.
Asking him open-ended questions such as "what are your thoughts about... do you wonder... (virtually any topic, such as: what happens next in a process, how stalactites and stalagmites formed this way, why people choose to eat different foods, how we could make this at home...etc)" may help him develop his own internal conversation about the things he is seeing and learning. These internal conversations can help him make connections between different ideas/concepts/facts and can also help with raising awareness of a budding curiosity on any particular topic.
Expressing your own curiosity to learn more about something, and role modeling strategies you utilize to satisfy your curiosity on that topic may help him as well. For example, this could be wondering aloud if you could make a dish at home that you enjoy at a restaurant... tasting it carefully and talking about what the ingredients may be, researching online for a recipe, and declaring some early attempts unsuccessful but worth another attempt at adjusting the recipe. Or this could be learning a new knitting stitch. Or learning about gardening, some element of home repair, or a new language, or a new vocabulary word of the day. Virtually anything which allows the child to see happy, well-adjusted adults as life-long learners with struggles and a desire to meet the challenge and try again.
DD5 seems to view the classroom as just one of several arena in which to show off.
There are some great books, such as Unwritten Rules of Friendship and others frequently mentioned on the forums which may help you teach him an approach which will serve him better in life.
I am thinking about signing him up for piano. He doesn't seem to have any special music ability. I'd like him to have to work at something.
This could be a great strategy, as long as he is supported in his struggles and failures... otherwise it may backfire and cause him to avoid anything new which he will not be instantly good at (or the best at). This avoidance is sometimes called "fixed mindset", and may be related to procrastination and perfectionism. The more flexible approach which accepts struggle and failure as learning opportunities is sometimes called "growth mindset".
