This usage of 'tout' is exactly as you imagined 'carrying'. Also with the notion of carrying with pride and/or 'obviously' / easily visible, noticeable to others. This meaning is not very common and in fact is not even listed in my Oxford Learner's Dictionary for example! Now "to tout" has a much more common meaning as 'to sell' - but it appears that this is not the meaning here but from the little context you have given it could be!

Touting means trying to sell something, usually aggressively. Example: "Jim was touting his merchandise" You may be thinking of "toting", which means carrying. Example: "He is always toting his skateboard"

It means that he possesses the skateboard and shows it off as a matter of pride and self-identity.