놓겠다는 is a short form of 놓겠다고 (말)하는. 놓겠다고 << 놓(다) + 겠 + 다고 말을 놓다 : talk in a casual[familiar] way 겠 : suffix used to express one’s intention 다고 : suffix used for indirect quotation 동갑이라며 is a short form of 동갑이라고 (말/주장)하며 이제 나나언니랑 동갑이라며 당분간은 말을 놓겠다는 민하 Minha who is saying that he will talk in a casual way with sister Nana for some time insisting that she and Nana are the same age now.

It is a united 어절(syntactic word), '놓겠다' (refer to 어간(stem) '놓겠-', but exactly apply '놓-') combined with <~(라)는> (간접인용 조사(indirectly quote postposition).

말을 놓다 means (s)he will use informal language instead of formal (honorific) language.