On the main homeschooling board I frequent, there is a forum for "accelerated learners" (read: gifted) and another forum for "learning challengers" (read: learning disabilities). It is not infrequent for the same person to post on both forums about the same child. You definitely have kids who are doing college-level calculus in fifth grade at the same time as struggling with dyslexia and an inability to write a coherent paragraph.
One of the things that we notice, once we've been on the boards for a while, is the frequent appearance of new homeschooling parents with kids in the preschool or early elementary years, who are looking for information for their "precocious" or "gifted" toddlers or little kids. Usually they define this as the child having started to read at 3, or can do addition at 4, or is reading a year or two ahead of grade level at K or 1. The child is, admittedly, doing well. But most people don't really define "gifted" or "precocious" that low. My first grader reads on about a third/fourth grade reading level. I think she's smart and she loves to read, but I wouldn't consider her "gifted." Gifted would be if she were reading AND understanding books at close to a high school level.
The main reason that I'd see in labeling a child "gifted" would be if you need special services/schooling for your child. A truly gifted child is generally not well-served by staying in a classroom based on their age. And skipping a year or two may not work, either, because they may not be able to socially handle being in a class with older kids, not to mention that not all gifted kids are gifted in all subjects. So having your child registered as "gifted" might be what you'd need to get the school to allow your child to, say, take classes at the middle school in the afternoon, while staying at the elementary school for the morning. Or getting in to certain special private schools or getting a private tutor. Or, if homeschooling, to help find other parents with children who are so far advanced, so that you can all share ideas and difficulties of parenting such a child.
The specific scenario given, though, makes me doubt the mom's label of her son as gifted. If she's labeling strange childish behavior as "gifted," then I'd doubt that she has any real idea of what "gifted" means. Yes, he might be gifted *and* weird, possibly because he's so far beyond other kids in intelligence that he has trouble figuring out how to relate to them (imagine yourself suddenly turned back into a 3-year-old... would you be able to join in activities and games with other 3-year-olds and have fun and not seem different?). But she's not doing him any favors by labeling activities that *aren't* gifted as being so. The little boy actually reminds me a bit of a daughter of a man dating a relative of mine. She was apparently a micro-preemie and adopted as a young child, and has a mix of ADD and asperger's, I believe. She acts in very similar ways to this boy. I think she's growing out of it a bit, though, as she's a good bit older than him now.