Hahaha. It can be quite embarrassing wishing someone a Happy Birthday on a wrong day but sometimes it does happen. Though the person may feel you do not care so much about them as they think.
There are also ways to cover up that mess by saying something else. You could be embarrassed if you say something like, "Uh, I thought it was your birthday".
You can simply say something like, "I got you, that was a joke on you, to see your reaction" or something like, "I am just practicing how I am gonna tell you happy birthday" or simply tell the truth by saying, "Because I lost my jotter where I wrote down all my important friends birthday, I am simply telling everyone a happy birthday so they can laugh and tell me the real one".
Don't worry about it, it's not the end of the world, if it's your friend or family member, he/she would understand. There's a saying around here that a Happy Birthday wish is valid for seven days :)
It's frustrating, i know and some people might get upset and blame you but come on, let's be real here, it happens to everyone, it's a honest mistake, no need to make a big deal out of it.
Just say sorry, and that should do it. We are humans, we forgive mistakes like this.
Just say "oops, my bad!" and carry on with your day. Don't make a big deal of it, and you'll be fine. Or you run with it, throw a party, and if someone says it's not their birthday, say, "who cares?" and keep on partying. Simple as that!
I would not overdramatize it. If you wish someone happy birthday in the first place it is a nice gesture. If it happens to be the wrong day, have a good laugh about that, say sorry and I think the person you wished well will not be cross with you :)