Just like "America" is a synecdoche for "The United States". :)
It depends on how the words are used; "Africa" to describe an area larger than a lot of continents combined, when you only mean a few countries - or just one! - is borderline offensive, though.
Using "America" to refer to the USA is muuuuuch less controversional and much less inaccurate than using "Europe" for all the countries in Europe, or "Africa" for all the countries in Africa.
For example, various US organisations have "America" or "American" in their names, and openly only deal with the USA (e.g. RIAA, MPAA, etc.). Organisations in EU that have "European" in their name are not focussed on just that country. People from the USA use the term "American" to refer to people from the USA, not all of the USA.
America/Americas is the entire land mass containing all North American countries and South American countries. It's just also slang for "The United States of America".
But there's a difference in that "America" or "the Americas" can include a much larger area than just the USA, while that's not true for the countries you list. China is agreed by both the PRC and the RoC to include Taiwan -- they just disagree over who is the legitimate government; "Germany" historically meant all German-speaking lands, but that usage is now, for understandable reasons, defunct; the Commonwealth of Australia is coterminous with the collection of islands collectively known as Australia; and "Ireland" is in fact the full official name (in English) of what is sometimes known as the Republic of Ireland to distinguish it from Northern Ireland, to which the former until recently laid claim.
Now, now (Brit living in Ireland here)... In fact it is the other way around. We laid claim to the island in 1801 forming the United Kingdom of Great Britain and Ireland. Around the turn of the 20th century, they weren't too happy with us though and wanted out, so in 1921 we agreed to let them have a chunk of it back (forming Ireland, Republic of and the United Kingdom of Great Britain and Northern Ireland). Now even today they still want the rest back, and half of the rest wants to go back (hence all the troubles in Northern Ireland), but we aren't letting them :D
I'm a Brit too, and I'm well aware of the history you describe. If anything, you've presented the truncated version: English/British interference in Ireland goes back more than 800 years, to the Anglo-Norman invasions of ~1170 and the award of the Lordship of Ireland to the King of England by Hadrian IV (the first, and so far the last, English pope).
I was alluding to Article 2 of the Constitution of Ireland of 1937, which explicitly claimed the whole island of Ireland as the state's national territory; this was replaced by more wishy-washy language about the Irish nation in the wake of the Good Friday agreement of 1999.
By the way, the Irish state created by the 1921 treaty explicitly wasn't a republic: it was a dominion of the British Empire, and the King of England was still technically King of Ireland (even under Irish law) until 1949.
> China is agreed by both the PRC and the RoC to include Taiwan -- they just disagree over who is the legitimate government
This is the historical position of the Taiwanese government, but it's decreasingly true amongst the Taiwanese people, so to some degree it's already an anachronism. Many actual Taiwanese don't self-identify as "Chinese" at all (while obviously recognizing the common heritage).
However, given the common tendency to use "Chinese" to refer to the greater cultural sphere, there can some confusion...
[I make this mistake routinely talking to people from Taiwan, saying something like "blah blah because you're Chinese" to which they immediately reply "No, no, I'm Taiwanese" (and this is generally amongst the politically apathetic young).]
In fact, I was led to understand that this is not a recent phenomenon, and that even before the flight of the Nationalist government to Taiwan, many (most?) Taiwanese did not consider themselves Chinese, until the Nationalist exiles started promoting that idea heavily.
Well take this with a grain of salt (it comes mostly from informal discussion with Taiwanese), but my understanding is that whereas there was originally a sort of ethnic/historical split to Taiwanese identity following the RoC flight to Taiwan, recent generations of Taiwanese are increasingly self-identifying as "Taiwanese" (as distinct from "Chinese") regardless of which side they come from ethnically, though the ethnic split is still something people are aware of.
I.e., many people now seem to have separate "ethnic", "cultural", and "national" identities (which can make conversations rather confusing).
It depends on how the words are used; "Africa" to describe an area larger than a lot of continents combined, when you only mean a few countries - or just one! - is borderline offensive, though.