I believe the confusion here stems from the usage of the term "Scotch-Irish" to denote the ethnic group in question. Scotch-Irish does not actually refer to a collective grouping of both Scottish and Irish peoples. Scotch-Irish is another (sometimes considered pejorative) name for Ulster Scots, which were a group of ethnic Scots who settled in Northern Ireland as part of an English state-sponsored plan to colonize Ireland and dispossess its landowners. So, in a simplified sense, Irish refers to ethnic Irish living in Ireland, Scottish refers to ethnic Scots living in Scotland, and Scotch-Irish (Ulster Scots) refers to ethnic Scots living in Northern Ireland. Source: Jane Kenyon, Jane Ohlmeyer, The Civil Wars: A Military History of England, Scotland and Ireland 1638–1660

Here's a reason from Canada, and a few others. For significant parts of Canadian history, "Gaelic" was what was used to refer to both the Irish language and Scots Gaelic. Between Cape Breton Gaelic speakers and Newfoundland Irish speakers it was the third most spoken language in the country, and "Gaelic", referring to both languages, apparently came to within a single vote of becoming an official language of debate in Parliament, given that there were a significant number of speakers. It was common enough one family story I was told is that one of Cree/Metis ancestors also spoke Gaelic.

Another reason why these communities get lumped together is that for hundreds of years, they literally spoke the same languages, and had a shared literary language, Old Irish, that lasted as a literary language for a very long time.

The strong separate identities we have today between Irish and Scottish stem at least in part to more recent cultural changes that impacted Scotland differently than Ireland - for example Scotland became largely Presbyterian, which became not just a church but at least a bit of a marker of identity against the British Anglicans, while Ireland developed at least some of its identity around the history of strong Protestant persecution at the hands of Cromwell and others, hence emphasized Catholicism. Although both identities pushed against British identities, they also put Irish and Scottish people against each other in a way that was new, and led to the situation we have today, where the "British" in Ireland are largely Scottish, who at one point were ethnically united with the Irish, but now are seen as 'the other guys' in a real meaningful way.

This still doesn't get into the differences between highlanders and lowlanders at all - just remember many of the highlanders are Catholic just like most Irish speakers, often with a similar cultural history that makes lumping them together fairly easy both from within, and also from the outside.