“Having a style” is synonymous with a variety of other positive things that new artists often struggle to achieve. Here are a few of those things:
Consistency. When you’re still getting a handle on your art, it often feels like you can’t draw anything the same way twice, or that your characters don’t look like they could ever coexist in the same universe. “Having your own style” here means you’ve finally learned how to duplicate your successes intentionally and that you can produce art that looks related without looking identical.
Of course, once you get to that point, you have to push yourself beyond mere consistency, but if you’re not there yet, it’s arguably a worthy goal.
Visual vocabulary. As mentioned in the original post, “style” is typically refers to an idiosyncratic visual shorthand; a triangle may be understood to be nose, or 25%M, 25%Y may be understood to approximate a white person’s skin color. This shorthand is best developed as the result of lengthy real-world observation, the distillation of real-world forms and colors into evocative imagery… but if art is your hobby, not your career, you might not have the time or the inclination to build your own visual vocabulary from scratch and instead attempt to synthesize a mishmash of other people’s shortcuts. For most artists, this experience is rather like being Voldemort and trying to use the Elder Wand despite failing to win its allegiance.
“Having my own style” in this case can be understood to mean “achieving some measure of comfort and familiarity with a visual vocabulary, ANY visual vocabulary, as long as it will actually work for me!”
Surpassing bland realism. Amateurish reliance on other artist’s shortcuts is only one possible cause of stiff and soulless art. Another major cause is getting so caught up on drawing from references that you’re still basically just copying someone else’s style ("someone else” in this case being reality itself) rather than “winning the allegiance” of the representational method you’ve chosen.
"Having their own style” means that at some point the artist must have been brave enough to depart from one-to-one accuracy, which is the only thing that allows artists to produce art that feels more real than reality does.
Brand recognition. This is the most mercenary reason to desire a distinctive style, and the most pressing reason that “tumblr famous” artists like poor viria so often get asked for advice on developing one: “having a style” facilitates recognizability and thus notoriety and popularity.
Not long ago someone requested art from me by saying "I’d love to see the Avengers in your style” and I panicked because I didn’t know what they meant – my representational drawings fluctuate wildly between realistic and cartoony, my lineart may be feather-light and sketchy or it may be thick and chunky, my approach to color encompasses everything from true monochrome to simple cell-shading to full 3D rendering – and yet the comment gave me warm fuzzies because it meant my art somehow stood out in that person’s memory as uniquely mine. I look forward to the day when my art is recognizable enough that members of whatever fandom or community I make art for can see previously unseen work in a fanvid or compilation post or whatever and go HEY THAT’S STARFIELDCANVAS! the same way the Les Mis fandom can most likely recognize art as belonging to prema-ja, princetenjolras, invisibleinnocence, littlewadoo, juanjoltaire, michellicopter, etc.
“Having your own style” is considered something of a prerequisite for standing out and rising above the masses. No wonder people want one.
Despite all the perfectly understandable reasons to want something that feels like “a style,” though, it’s still not something you can really force. You know how when you were eight years old and signing your name on the back of your first library card, you intentionally tried to make your signature look smooth and mysterious like your parents’ signatures and it just looked like you had terrible cursive and possibly Parkinson’s, but now that you’ve been signing receipts for half a decade your signature is a deeply personal linkup of fluid loops? It’s like that. The only way out is through.