In general, research and education is one big (read: global) network. Unlike it may be the case with commercial entities, competition usually happens on friendly terms (you try to be better, but your goal is not to make the competitor disappear) and it does not preclude collaboration.
A few members of one university might get caught up in feelings of rivalry, but I cannot imagine that to have any real effects towards people with links to the so-called "rival" university. We're talking about universities, not elementary schools.
With this in mind, any obvious links to other places (e.g. clothing from another university) seem positive; they underline that the lecturer has gathered some experience at various places, and thereby, indirectly, imply that the university has a well-connected staff with diverse backgrounds.
In the described situation, one thing to note is, however, that there should be some connection to the institution advertised on the clothes. As described above, when I see staff at my university with clothes advertising a different university, I would consider that positive (=> "people with experience and connections"), but if I found out they have not actually been to that place, I might feel a bit cheated, even though I wouldn't care otherwise.