For example, your app icon would be blue and black and your logo on printed shirts would be red and black.
Building a brand takes more than a logo. With that said, consistency is key for obtaining a competitive advantage that speaks to your market for longer. I would recommend against using different styles and colors for various purposes and instead maybe avoid using in lieu of the logo use maybe instead borders or patterns that use your logo's or brand colors. The idea of a logo is to engrave a mission or product into potential customers when they simply see the brand or logo... Once a logo is pushed and promoted you can strengthen that image by enforcing the brands colors through different materials or media :)