This is an interesting question, and I think it can't be answered in general, as people have different opinions on this.
For one thing, there
is a sort of Highland/Hebridean ethnicity, many people there look different than in the rest of Scotland and Britain. When my husband came with me to the Western Isles for the first time, he said, "hey, lots of women here look like you" - meaning, dark-haired, pale-skinned, blue-eyed, slender.
Now, that's not directly connected to Gaelic, I suppose it's rather the result of a very specific mixture of ethnic groups which have been settling those areas for the past few thousands of years
You will find lots of people in the Highlands and Islands who do not consider themselves anything but "British". Now, that's not really an ethnicity, it's a rather pragmatic national identity. Many elderly Gaels (or people with a Gaelic family background) are sort of ashamed of their Gaelic roots, because for them "Gael" means something like "a clueless person from a remote islands who had difficulties adapting to the mainstream society when trying to earn a living in a place like Glasgow". Others are quite self-conscious about being Gaelic, but I think most see it as a cultural rather than an ethnic thing.
Then you get those people from the USA who are awfully proud of their "Gaelic ancestors" and who are completely convinced of their own Gaelicness (which entitles them, among other things, to immediate tattoo translation help on internet forums). That's something many people in Scotland can't really understand, although there is a strong sense of heritage, genealogy, local community, etc. especially in the traditionally Gaelic-speaking areas.
So, it's a complex topic, because ethnicity, language, culture, and "bureaucratic" sense of identity are not necessarily consistent with each other. This is not a specifically Gaelic thing, though, it applies to most people in Europe, I'd say.
Personally, I know that my ethnicity is a totally mixed up thing, and I think it doesn't really mean much to me in terms of identity. Culturally, I feel neither German nor British. I do relate to the regional/local culture of the Rheinland where I have spent the biggest part of my life so far, and to the Gaelic culture of the Hebrides as well.