On the other hand, he was, as you admit, born in Scotland, spent many years of his childhood there, including (I may be wrong) the formative secondary school years (ages 13-18), and I don't think it would be crossing the line if some proud Scot were to claim him as one of their own.
Your second line vaguely defines a 'people' but given the context, I am assuming you suspect me as one of them, so for the record, I am English-born of a largely Scottish origin family, foolishly proud of my Scottish heritage, and do not nurse any grievance - quite the contrary. Indeed, I voted for both recent 'Scottish' prime ministers. I've never noticed a grievance in others claiming Blair as Scottish but will look out for it.
Your last sentence makes a sad but good point.