I'm just going to dump some of my research notes here, as well as my interpretation of the imprint design. My notes are totally rambly, but maybe they'll prevent you from (or inspire you to?) go down some of the same rabbit holes, haha:
1) The image that's repeated (upsidedown and rightside up?) inside the swirl/S shape really does remind me of a "knockoff" Quiksilver logo -- a wave on the left over a mountain on the right, with a blank area that has three little "arms." Quiksilver has made some products with stylized versions of their logo, but I didn't find anything similar to the image I'm getting from the print. They do even sometimes use a double-version of the logo with a similar orientation:
NWOT QUIKSILVER 100% cotton white short sleeve with double logo in neon size L | eBay (I believe the Quiksilver logo is based on the Waves Over Kanagawa painting, so I searched for other logos inspired by that image, as well as a vast combination of "wave/mountain/etc" keywords in the US Trademark catalog under the apparel and shoes category, and didn't turn anything up.)
2) I considered what the logo was meant to represent -- ideally it should be representative of a company in some way, right? The "mountain" also resembles a skateboarding quarter-pipe, and a lot of others here have gone in the skateboarding direction here, too, for a few reasons. That might make sense if the juvenile is a teen, and as someone else mentioned, it doesn't make sense to have the smoothest area of tread on the ball of the foot -- EXCEPT when skateboarding that can be beneficial. Still, I didn't have much luck going down that route either.
3) I tried image searching "ACA," "BCA," "CCA," "DCA," etc with shoe and sole keywords. I turned up a few hits but none of them really seemed to pan out:
***GCA is a term used by shoe enthusiasts to mean "Greater China Area," indicating a knockoff. These are usually decent knockoffs of Nike and Adidas shoes, though. (Also why would you indicate on your shoe that it's a knockoff, haha?)
***MCA is a variety of Nike Airforce 1s (again, I'm thinking knockoff here, obviously the actual Airforce 1 soles don't match this) -- apparently this stands for Museum of Contemporary Art, who Nike partnered with for the release of this shoe. However, this shoe only came out in 2019.
***OCA is a style of shoe by a brand called Cariuma -- nothing here seemed relevant though
***RCA and TCA are both models of New Balance Shoes
4) Since nothing panned out with any "CA combinations, it got me wondering whether "CA" could actually be "EA" -- there's potentially a hint of what could be the middle horizontal bar in the C. I took lots of liberties with the blurrier letters and tried a few things like "SEA - LAND" (could be a reference to the interpretation of the Quiksilver logo? or amphibious/water shoes?), "ZEALAND," etc. I still didn't come up with any promising hits though. I also tried interpreting the second word and found a company called ZEAZOO, one called ZAQQ, another Zao, etc. but no real hits. The Zao "company" looks pretty shady, and they sell knockoff Versace shoes, whose swirly logo on the sole is very vaguely reminiscent of the pattern on the imprint, but it's a stretch).
5) The only other sole I found with similar plain text (i.e. an unembellished "Arial"-like font in a straight line, not a logo) marking was this one:
only other shoe with plain brand printing on ball of foot:
Espadrilles Flat Rope Soles for Woman Men and Kids Made - Etsy ... which appears to be manufactured by
Espadrilles soles and this got me looking into sole manufacturers, primarily in India and China, like these:
Shoes Materials & Accessories directory - updated information on Shoes Materials & Accessories manufacturers and suppliers from across the globe
www.tradeindia.com
Shoe Upper, PVC Shoes Sole & PVC Slipper Sole Manufacturer offered by K.B Industries from New Delhi, Delhi, India
www.indiamart.com
6) Finally, Vibram soles have a variety of "three-leafed" shapes on their soles (the clover in the imprint really looks like three hexagons together to me, but I could be totally wrong there). The plain text doesn't seem like something you'd ever see on a Vibram sole though. Example of the "clover" here:
Vibram MULTILAND 966C Shoe Sole Manufacturing Boots Shoe - Etsy India
It would be great to have a better idea of the date the imprint was left, so we could rule out newer shoes. Was it closer to a year ago, or 10 years ago? Also some location information might help -- country at the least.
