I'm one of these anti-logo folks (a severe case, I dislike text as an aesthetic and won't get anything with writing or symbols). I know, you're thinking "how can I find clothes without all that stuff?" And I'm here to tell you, it's simple: there are fewer choices to consider. See a rack full of a brand that thinks its logo is more appealing than your personal brand? Keep moving. Less choices, less option paralysis, less time shopping.

With men's clothes, you can buy plain shirts, socks, etc in 3-packs (there's few choices, but few are needed). Women's clothes tend offer more unbranded options, unless you're buying athletic wear or a purse. Casual shoes generally suck in this regard, for men and women.

This has been getting more difficult, of late. It used to be that logos were embroidered on tags or patches, which were then stitched onto the clothes. But with embroidery robots and the rise of counterfeiting, companies are making it more difficult to remove their brands.

edit: but practically? Thrift shops have a wealth of logo-free stuff

As an alternative, sew your own patch over it. There are plenty to choose from in craft stores, on Etsy, even department stores probably have a few inoffensive ones, if they have sewing supplies.

A blank patch looks sort of conspicuous, but a cute little non-corporate owl or fox will look like it's meant to be there.