Not long ago I had a fellow dudedad send me a funny picture, capturing a moment of desperation in the backcountry. His group's beer stash had exploded and they were salvaging the remains. He happens to have his dudedad hat on at the time, but noted it probably "wasn't dudedad stuff."
But my first thought was, hell yes it is!
Here's the thing; when you're not with your kids and you happen to have the dudedad logo on, or in your pocket, etc. you're going to be more conscious of your most important job: dad. It raises your awareness.
This isn't something we intended, but it is a factual-byproduct, and we can state that based on experience. While the clothes don't make the man, as they say, wearing or carrying dudedad stuff raises your game. You're always in dad-mode, on some level.
So, even when you're out doing dude things with your friends, you're still a dudedad. It's a state of mind...and a lifestyle. The stuff is just something of a reminder.