Scots meant scotti, meaning Irish settlers from Dal Riata originally. Basically all Gaelic speakers were referred to scotti. Circa 1300s in the Bruce invasion of Ireland they began calling themselves king of scots to gain some solidarity with the Irish against the english as Ireland and the Irish lacked a proper king like figure. However the Bruce invasion was disastrous.

Also important to note the idea of Gaelic/scot/Irish solidarity is one with deep roots. For example the Christian clergy in Ireland circa 1010s crowned Brian boru, high king of Ireland, imperator scottorum literally meaning emperor of scots. The king of alba even sent him troops, some historians say as tribute, to fight against the Norse in 1014 for the battle of clontarf.

However as Irish became out of fashion and English became dominant they became more bent, especially in the south to wrestle off their Irish heritage, going so far as to get a papal decree declaring exclusive rights over the "scots" name to those in Scotland exclusively.

For clarification, are you asking why the title is 'King of the Scots' and not 'King of Scotland' akin to why Napoleon was 'Emperor of the French' and not 'Emperor of France'?