I think the answer is
I think the title refers to
Rewind or "Re: wind", "Re" being an abbreviation for "in re" meaning "in response to" implying a subject about which you wish to speak.
When you say that I'm feathery, that's quite clearly wrong.
Could refer to "downwind" not being feathery (credit to deepthought in the comments)
If I come from the trees then you can join me in song.
Referring to woodwind musical instruments
I've nothing to wag, but you imply that I do.
A tailwind is one that comes from behind.
I have nothing to sell, but you suggest that's untrue.
Referring to "trade" winds (thanks to deepthought for pointing this out)
You say I'm unwell if I give you no prize.
"It's an ill wind that blows nobody any good"
If you ensnare me I'll bring tears to your eyes.
Trapped wind can make you cry
You say that I'm angry when I turn to the side.
A "cross" wind (crosswind) comes from the side
But call me divine and you'd better all hide.
The Japanese word Kamikaze translates as "divine wind"
Thanks to the commenters for their help especially deepthought who sorted out two of the lines.