C'est ma chambre à coucher.

English Translation

It is my bedroom.

Would it not make more sense to use “dormir” here rather than “coucher”?

“Une chambre à coucher” is idiomatic for “bedroom”. You’re right that in general you’d avoid the verb “coucher” in modern French because it’s usually used to mean “have sex with” (like the famous “voulez-vous coucher avec moi”). “Chambre à coucher” does not carry such an innuendo however.