Interesting question. I think a lot of it comes from the ongoing confusion/interchangeable use of the terms ‘Great Britain’ and ‘UK’, when they’re not really the same thing at all. You’re right that it wouldn’t break up Britain, but it would break up the UK – and for some people, that means the same thing even if they don’t realise they’re technically wrong. That said, I don’t think the sense of “we wouldn’t be who we are now anymore” that some people find reason enough to be in the “no” camp should be discounted, even if it is pretty vague and nebulous when it comes to reasoning and they’re talking about the UK but saying Britain.
However, the British cultural identity question is also a double-edged sword. Because if Scotland leaving the UK as is can be seen as having no real detractions to British cultural identity, or at least no detractions that aren’t largely trivial, then the natural counterargument is what would Scotland then gain from independence on a cultural identity basis? If British identity is not under threat from Scotland not being in the UK, then there is inherently nothing to be gained by Scotland from leaving the UK in this regard – but I’d be willing to bet that a lot of “yes” campaign propaganda will trumpet freedom of cultural identity as a key reason for leaving. And if that means Scottish people will then be encouraged to see themselves as primarily Scottish and not British, and a yes vote succeeds, then something *has* been lost from British identity. You can’t take something away and end up with the same thing, and British identity will change if this happens.
So I’d argue either there’s no benefit for Scotland leaving, in which case this particular avenue re: independence is largely moot. OR, Scotland leaving would detract from British identity because British identity would largely lose the influence of the Scots, at least in the short term while Scotland was (quite rightly) busy asserting itself as a new country. I think that’d be a loss, for one.
That said, as a disclaimer I would be entirely in favour of an independent Scotland as long as it was not to the detriment of the country – my main concern would be that the people in power pushing for it are doing it for ideological reasons. My only other concern is really the scenario where both Scotland *and* the remaining UK would suffer by both being diminished, (i.e. the whole as it is being greater than the sum of its parts). But of course there’s no way of knowing if that would be the case in advance.