Why would the character mention "dry clean" here?
Native speaker’s answer
Rebecca
The speaker is making a joke here! Normally the tag or washing instructions of a clothing item will be on the inside, worn at the back or side. The speaker noticed that the tag was at the front, near her bust. Thus, she says "so the designer's name isn't dry clean only" to show that she has realized she has worn the dress incorrectly and has worn it backwards instead.