Is a father of a newborn baby allowed to wear the baby ID bracelet in a place where is no eiruv? According to Rav Moshe it would seem to be muttar as it is worn on the body itself. Although he writes the bnei torah should not wear watches due to the fact that some people might think that carrying a pocket watch is okay, he holds that m’ikar hadin one could wear a watch as it is worn on the guf h’adom. (Also that was written around 60 years ago. Today I don’t know of anyone who uses a pocket watch. I am not sure Reb Moshe would be machmir on a baby ID and have the husband stay inside the whole shabbos and miss davening with a minyan based on a gezerah on something else that doesn’t apply today).
Also, could it be considered a tashchit? A person wants to show off to his friends or to the gabbai in shul that he had a baby. Would it be similiar to a chosson wearing his chosson watch, he would wear it even if it broke to show to other? It is something choshuv that he would want to show to others.
The father of a baby or anyone else with a Bracelet ID should not walk out with it on shabbos. Although R’ Moshe does say that something that is touching the body is permitted to wear outside on shabbos, however this is only regarding something that is normally worn, but not with an ID bracelet that is only temporary,( see sources).
Regarding the bracelet being a tachshit, it is not similar to a choson wearing his choson watch, which is something that is always considered a tachshit. The ID bracelet is not meant to be beautiful or to decorate the person wearing it, and even the proud father that is all proud of his ID bracelet, to show off that he is now a father, however in two weeks from now he wil e quite embarrassed to be seen that way. It may however be worn if it will be used to close the cuff of the persons shirt. this can be done by opening the sleeve button, and sticking the sleeve under the band, so that it is now serving as an accessory to the persons shirt.
Although R’ Moshe zt”l wrote in Igros Moshe O:CH 1-111, that it is permitted to wear a watch on shabbos because it is considered an article of clothing, however it is very difficult to understand. See Megillas Sefer Shabbos 41-2 that asks numerous questions on this idea. First of all see 301-12 that a zav may not walk out with his bag, even though it is touching his body. Also see O:CH 301- 20 that we can’t walk outside with a mask etc, even if it is touching the body. (See Igros Moshe Y:D 2-47 that a woman may not go outside with a moch dachuk). Besides everyone will agree that one may not take whatever they want outside, by merely taping it to his body (before shabbos). Therefore it would seem that R’ Moshe means that the watch is permitted because it is normally worn that way and that it touching the body. See Shut V’dibardo Bom siman 88, that R’ Dovid Feinstein shlit”a said, that the teshuva is not clear enough, (due to the many questions on it), and that his father meant that a watch is permitted because it is normally worn and it is touching the body.
In Shabbos Kehilchoso 40-32 it is written that one should not wear an ID bracelet outside in the street on shabbos.