There are 2 possible explanations:
1) The en is a pronoun that means "because of that, on basis of what was said before*. J'en ai le droit d'aimer means then "I have the right to love because of all of this*.
A less literary and more mundane form would be: en raison de tout ce que je viens de citer, j'ai le droit d'aimer.
2) the en is a pronoun that replaces d'aimer. The syntax can be surprising as the pronoun precedes the word it replaces, and as the referent is then repeated and seems redundant. It is a turn of phrase that can be found in the flow of a conversation in spoken French.
J'en ai beaucoup, des livres.
Il m'en a parlé, de sa soeur.
Note that in a written transcription, it is then appropriate to add a comma before the repetition of the referent, which I do not see in transcripts of the lyrics of the song (but...I have seen many incorrect lyrics transcripts on the web).
It has even become the standard form for some fixed phrases like en avoir marre de, en avoir assez de, etc.
Another element you should keep in mind whatever interpretation you choose: for song lyrics, there is also the constraint of euphony and of the number of feet that needs to match the music. Maybe the lyricist just felt j'en ai le droit d'aimer sounded better on this melody that the standard j'ai le droit d'aimer and simply used his "poetic licence".