The dog’s uniform shows The Little Red Book…
… poking out of the left front chest pocket.
The breed appears to be Pekinese (several examples pictured at this entry)…
The breed was favored by royalty of the Chinese Imperial court as both a lap dog and companion dog, and its name refers to the city of Peking (Beijing) where the Forbidden City is located.
So part of the not entirely sly in-joke is that preferred dog of the Imperial insiders (i.e. “class enemies”) is part of the Red Army.
My guess is that these visual references make the subject matter Chinese, no matter where the towel was made and printed.
ETA: grammar, clarity, bedtime for me clearly (second Pfizer COVID shot today makin’ me tired )