Is a man allowed to wear a "dog tag" type of thing? Is it beged eesha?
Most forms of "beged isha" (the prohibition on men wearing women's clothing) all depend on a given society's norms of what's called "men's clothing" vs "women's clothing." (E.g. a kilt is okay in places where men wear them!) While certain actions or categories may be objectively off-limits as "beged isha", I've never heard anything about a necklace-like thing being one of them. (Didn't people use to wear money pouches around their necks? Or better yet, in the Gemara they had amulets that you wore around your neck -- for men or women.) If armies have been using dog tags for all their men for (how long has it been?), it's hard to call this a "feminine practice."
Even if it were, I don't recall the sources but the conclusion on this topic I always heard was that if it's still clear that you're a man, to wear one item of women's clothing for some functional reason (non-decorative) is permissible -- e.g. stuck when it's pouring and all you have is a woman's rainhat, yet no one will mistake you for a woman; Rabbi Frand gives the example of a bearded man in a normal suit, wearing a sheitel just for the sake of Purim silliness. The same should apply here.
But as always, ask your rabbi.
"Dog tags" are worn primarily, and until recently almost exclusively, by men. It would be very difficult to justify labeling them a womens garment.
A more likely problem is that when one wears them outside of their intended purpose, i.e. as a style rather than to to identify a soldier, it may be similar to wearing a sword which may present a problem of Chukas Hagoy.
Regarding the leniency of some permitting a single garment many poskim disagree and many/most illustrate the prohibition by cases where the person is not entirely dressed as the opposite gender. I would remind people that this is an issur d'Oraisa and not to be quick to follow lenient opinions without consulting their Rav.