For example, your app icon would be blue and black and your logo on printed shirts would be red and black.
So, I think the thrust of the answers are: "No"... Unless you have a good reason to do so.
There seems to be no real need. All the reasons given by other contributors I would pretty much agree with are (possible) valid reasons - different countries, different product groups, different channels etc
But you're asking.... App Icon... T-Shirt Icon...
So, brutally, I wouldn't waste another minute of your precious brain power on it unless you have a tangible reason that is CUSTOMER focused and makes it a very valid reason to do so. Otherwise you are just distracting yourself from your core business activity with meaningless design (art) activities and neglecting your business.
