Memes are probably more used than rorschach tests nowadays, but that's because the tests are outdated and compromised by everyone knowing about them. They are also dubiously useful to begin with, as they don't completely avoid the issue of having the therapist guide the patient's answers. Memes would be even more problematic in guiding answers.

Memes and social media already factor into studies, but I doubt they'll ever be a therapeutic tool. They could never be up to date, and they often rely on complex cultural connections to generate their humor. At most, we'll see funny comics or captioned images used, but they'd never be like wild internet memes.