I think it might be a pun as well. To pun is to play on the multiple meanings of a word. This paradox contains several applicable and different possible meanings of the word "die":
Meaning 1: dead, no longer alive
Meaning 2: fade away
Meaning 3: to faint or languish
Meaning 4: lose force, strength or active qualities
Finally, the expression "never say die", evoked by this paradox, means to give up hope, abandon effort or surrender.
Cowards most certainly do this many times before they die. So, yes, very punny.
An interesting question. I would say no. A paradox is a conceptual contradiction—cowards can't literally die many times before their death. It isn't possible. However, the line feels true, because of how they suffer. A pun, by contrast, swaps words that sound the same but sound different (like "grave" [serious] and "grave" [a hole where you bury a body]) in the literary terms page's definition of pun. Puns involve substitutions of language and depend on fooling the ear for their humor.
So, the two concepts are related, but unless you hear a specific word swap in that line, it is a paradox but not a pun.
Pun entails word play. Here, it is not found. A famous pun is Hamlet's line in Shakespeare's tragedy "I am too much i' th' sun". The pun is on the words "sun/son".