Designs and sayings attached to velcro or sewn directly to something are commonly referred to as morale patches. In the military patches are usually mundane - names tapes, Branch identifier, Unit Patch, US Flag, blood types, etc. Morale patches are a way of adding something a little different to your gear. Rarely worn directly visible on the uniform, but often on other gear these nonstandard patches are a fairly simple way to make a statement or offer yourself and others a little motivation.
I used to wear one of these on underside of my collar and gave one to all my guys, it was a simple and subtle way to build some team/squad camaraderie...
This is a one I have on my EDC backpack pictured with another patch... These are meant to remind me to always be exploring and finding new things...
Here are a few that ride on my key chain, the larger one is a Viking magical stave known as "Lasabrjotur" It is meant to enable the carrier to gain access to any lock, with it are 2 small company logos... The two companies make some pretty cool and useful little bits and bobs....
Next up are two of my favorite patches, the one of top is one I earned, if you like hard physical endurance events check out Goruck events. The bottom one is a take on a business card from one of my favorite old Western TV shows, "Have Gun - Will Travel"
Overall morale patches don't have a huge purpose other than raising spirits and providing a bit of motivation!
If you like these I've got many more I'd be happy to share!