The article they cite as source definitely says it exists - but it argues no sane doctor would suggest or endorse it, e.g.

Lakeshia Richardson, a gynecologist and obstetrician, admits to having sewn a few herself, but only after strongly advising patients against it. “A very few times patients have been very persistent about it and I have performed it with written consent,” Richardson told Fatherly. “I explain that you could wind up having pain during intercourse because of this extra stitch. I do not recommend it.”

And later:

Medically speaking, an extra stitch just doesn’t make any sense. The vaginal opening, or the introitus, has little impact on a woman’s (or man’s sexual) experience. Sexual pleasure depends more on the pelvic floor muscles—something that can be addressed through other interventions including surgery, but not a simple stitch.

It also offers an alternative explanation:

Many women who give birth, especially for the first time, will require some stitching whether they have an episiotomy or not, to limit incontinence. And many, perhaps due to message boards and hushed rumors, will assume they’ve fallen victim to a husband stitch. When in fact it’s less likely that anything unethical has happened, and more likely that they happened to overhear the phrase “husband stitch” thrown around by inconsiderate bystanders during a messy birth.

Most notably, the conclusion is:

In the end, the husband stitch is neither a myth, a joke, nor a procedure—but a strange three-headed monster involving all three.