Good question.
浴びる is a 再帰動詞 "reflexive verb". 再帰動詞 takes object(ヲ格補語), so it's transitive in this sense. However, 再帰動詞 ALWAYS affects only the subject themselves, i.e., both the agent and patient are always the subject. Also, it doesn't have the passive voice except so-called 迷惑の受身 "passive of adversity", so it's close to intransitive verb semantically. 着る and 脱ぐ belong to 再帰動詞, too. As same as 浴びる, these verbs don't have their intransitive counterparts, instead, they have true transitive counterparts 着せる and 脱がす, respectively.