Common metaphors
Spring / Winter
Spring represents reuniting with someone you miss, after a long time. In this context, winter represents the period you're separated from someone and missing them. This metaphor was first explored in depth in Spring Day.
Wings / Flying
Growing wings and flying represents achieving your dreams. The imagery largely began in the Wings era. More recent songs often mention ARMY giving BTS wings (enabling BTS to manifest their dreams) and BTS flying higher and higher (gaining greater global success).
Sea
The sea represents BTS's dreams coming to fruition, in contrast to being in the desert when they were at risk of failing as a k-pop group. One notable reference is in Black Swan. They discuss being at the bottom of the sea, meaning they're successful but feel emotionally disconnected from music (my interpretation). The sea doesn't always represent their success, as an earlier love song uses it to represent a love interest's affection.
Hero
BTS is ARMY's hero in Anpanman, which talks about how BTS doesn't have special powers but is committed to helping ARMY whenever they need it. (In press and in lyrics, BTS has expressed a broader sentiment of 'if we're not the ones providing fans with comfort, hope and happiness through our work, who else is going to do it?')
Galaxy
BTS songs have always included a lot of imagery about galaxies, stars and planets. In more recent albums, the galaxy represents ARMY. This was notably the central metaphor in Mikrokosmos, which is about how BTS and ARMY are individual mini-universes that make up one big universe.
Rain
Though rain is a common literary metaphor, it is also the central theme of the song Rain (Dark & Wild album) and mentioned in several other songs to express a prolonged feeling of melancholy or hardship.
Crow-tit
Crow-tit is a small bird that represents underdogs or those coming from less privilege, often with the subtext in BTS songs that they were able to succeed in spite of this unfair system. As mentioned in the song Silver Spoon, crow-tit is contrasted with storks, who are the privileged members of society and in a sense, the oppressors who uphold this unfair system. Based on my understanding, this metaphor of the crow-tit is derived from a Korean proverb.
References that indicate BTS's growth and maturation
In addition to metaphors, BTS also echoes past lyrics using similar sentence or verse structure, but changed to an opposite meaning to highlight growth and progression. Most of these connections are fairly common knowledge / official interpretations.
Best of Me / Magic Shop
Best of Me talks about loving someone so much that you give them the best of you. Magic Shop inspires you to take care of yourself, as they say "You gave me the best of me / So you'll give you the best of you".
Boy in Luv / Boy with Luv
Boy in Luv is about obsessive love and Boy with Luv conveys a calmer love that is able to appreciate the little things about someone.
N.O / ON
N.O and No More Dream both talk about rejecting societal pressures to take standard paths of success and to insist on pursuing your dreams instead. ON is about enduring the challenges and pain that come with achieving their dreams, and to insist on continuing to live their dream anyway.
BTS Cypher Pt.1 / BTS Cypher 4
The Cypher series are general responses to people who try to put them down. As BTS has gotten more successful, we can see them feeling more secure and softening their anger between the first and the last Cypher. While Cypher Pt.1 says "I’m going to cut you one by one", Cypher 4 says "[The insults are] making me bored". Even the pace and general sound of Cypher 4 is calmer.
Save Me / I'm Fine
Save Me is about desperately needing someone's love. I'm Fine is about being fine without it. Each verse of I'm Fine corresponds to a verse in Save Me, but saying the opposite.
• Save Me: "Gimme your hand, save me save me"
• I'm Fine: "I'll let go of your hand now"