Have any cats had problems after drinking mouthwash?
Cats haven't had any problems drinking human mouthwash. Normally, human mouthwash is safe for cats, but only for rinsing or external use; it's not suitable for internal use. This is because mouthwash is produced according to certain hygiene and safety standards. If it's safe for humans to use, it's certainly safe for cats to use as a mouthwash. Human hygiene standards are definitely higher than those for animals. 
The function of mouthwash
Mouthwash has the effects of cleaning the mouth, reducing plaque, inhibiting bacteria, freshening breath, reducing tooth decay, and treating oral diseases such as periodontal disease, dry mouth, and oral mucosal diseases.
Rinsing with mouthwash not only prevents bad breath and makes the mouth clean and comfortable, but it also washes away food debris that hasn't yet been fermented by bacteria from the surface of teeth after meals, reducing the chance of dental diseases. The correct way to rinse is to hold the mouthwash in your mouth, close your lips tightly, slightly separate your upper and lower teeth, and move the muscles of your lips and cheeks to allow the mouthwash to fully contact the tooth surface, gums, and mucous membrane surface. Use your tongue to move the mouthwash back and forth between your teeth to achieve the desired rinsing effect.