Is it significantly inappropriate to wear clothes bearing the name of University X, when working as an official employee at University Y?

There are definitely cases in which it would stand out as strange or inappropriate. You should be careful in cases of a rivalry, since that may be viewed as a significant statement even if you meant nothing by it. (I'm thinking of Harvard vs. Yale or Illinois vs. Iowa. Athletic rivalries are particularly troublesome, since students may consider the clothing a sign of active support for the rival university. No reasonable person could take offense, but not everyone is reasonable.)

It's also risky spending too much time wearing clothing emblazoned with a much more prestigious name than the university you are currently at. You don't want to look like you are ostentatiously reminding everyone that you graduated from Harvard, unlike the riffraff around you.

I don't think either of these will be an issue for wearing a Dalhousie University hoodie at Nankai University. If this were in the U.S. or Canada, I don't think it would be a problem at all: the worst case scenario is occasional good-natured teasing, and even that's far from certain. However, I have no experience with how it might be viewed in China, so it's worth looking into that specifically if you have reason to fear the reactions might be different.

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.

If there are hot rivalries between the two universities, then it might be taken as offensive or rude.

I am not familiar with the universities you mentioned, but if you were talking about Texas Tech vs. TAMU or Oklahoma University vs. Oklahoma State then I would say definitely wear the local colors.

Further, as a faculty member - you are almost expected to promote the university brand; and it might seem as odd that a faculty member is wearing "opposing" colors - again, the degree of animosity you would face would depend on the degree of competition between the institutions.