> I went to Ireland and visited a beach. I took home a rock as an souvenir of my trip. I live in America, and I wonder, are the rocks in Ireland different from American rocks?

It depends on where you were in Ireland and, since you're dealing with a rock you picked up from a deposit of loose sediment, where in Ireland the particular rock you picked up came from. Like most places, Ireland has a variety of rocks of different ages and geologic histories that are exposed (e.g., the wiki page on the Geology of Ireland). Depending on where in Ireland your rock came from (and where in America you're comparing it to, as similarly, there are lots of different ages, compositions, and histories of rocks in North America), it might be quite different or it might be pretty much the same. For example, portions of what now is North America (i.e., Laurentia) and Ireland (i.e., Avalonia) were joined during the Caledonian Orogeny. There are formations (i.e., packages of rocks of the same age and similar lithology which represent deposition in a single largely continuous depositional environment in a particular area) which appear in New England and Ireland (and elsewhere), e.g., the Old Red Sandstone.

At a more basic (and slightly pedantic) level, it also depends on how specific you want to be and how you want to define "the same." Let's consider a sandstone from Ireland and one from America. If you keep it super shallow and stick with basic rock type, then for sure you can find the "same" rocks all over the world, i.e., a sandstone is a sandstone regardless of location. As you get more specific, things will narrow as you consider differences in grain size (e.g., coarse sandstones are not the same as medium sandstones) or details of composition (e.g., arkoses are not the same as quartz arenites). If you keep getting more specific, you could start considering do these two sandstones have the same age and source, i.e., do they represent effectively the same original deposit. All of these definitions of two rocks being the same would be appropriate depending on the context. For this last, most strict type of "sameness", let's consider the other part of your question.

> Like if I ran science tests, could the rock be determine it's not from America. Or all rocks, just rocks.

For simplicity, I'll keep my focus on sedimentary rocks, and then this would broadly fall under the umbrella of sediment provenance. There are a wide variety of characteristics of a sedimentary rocks (e.g., age populations of detrital minerals, sand composition, and a variety of geochemical proxies) that can be used to "fingerprint" the source of these rocks and thus establish individual sedimentary rocks as semi unique. Being able to identify a particular sedimentary rock through provenance analysis depends on the details of the rock in question, i.e., some may be more unique than others and easier to fingerprint. Because of plate tectonics, most landmasses have complicated histories where currently disparate bits may have been in close proximity (like the example with the Caledonian orogeny and the Old Red Sandstone), so it's not as though all Irish rocks will be distinctively Irish or something, but rather rocks of different ages will reflect the plate tectonic configurations and depositional environments at the time they were formed and thus will have "affinities" with rocks in different places.