It seems there is a lot of confusion about various military and veterans discounts, and the identification that will or will not qualify you for the discounts. I'm no expert on this subject, but I'll tell you what I do know. First, there is no "veterans discount card." There are a variety of different forms of ID, including a military identification card for active duty, National Guard, reserve, retiree and family members. If you are eligible for care from the Department of Veterans Affairs (VA), you will receive a Veteran Identification Card (VIC). A VIC may be noted to indicate that the eligibility is service connect, that the bearer has received a Purple Heart, or that the bearer is a former POW. There are also membership cards issued by various veteran's groups, including the American Legion and the Veterans of Foreign Wars. Many people use their DD214 for identification. I've even discovered a few shady companies that are selling cards that are designed to look official, but are just the design of that company and a way for them to make a buck off of a veteran.
None of these identifications automatically entitle you to any discounts or privileges. Each individual business determines its own criteria for offering a discount, and each individual business determines to whom it is going to offer a discount. Some generous companies offer discounts to all military and veterans, and other require that you fit into a certain category in order to receive a discount. It never hurts to ask, but remember - these companies do not have to offer a discount, and if everyone complains about it, they'll probably stop!
If anyone else has other useful information to add in the comments, please do. We can all learn together.