Do I need to backup my mini iPad if I have already replaced it with another iPad?

Assuming your transfer was totally successful, I don't think so.

Check and re-check if everything is okay on the new iPad, then start creating a backup of that new device on a regular basis.

Also, do not forget to perform this procedure on the old iPad, before returning it for your trade in:--> What to do before you sell, give away, or trade in your iPhone, iPad, or iPod touch - Apple Support

Regards

Giulio