I've noticed Sen. McCain saying, "I come from a long line of McCains who have served this country" quite regularly recently.

The intent, I believe, is to emphasize ancestry over character or to suggest that ancestry is character: like those tired old 19th century novels that pitted the noble aristocracy against the crass nouveau riche. The Senator sees himself as the patrician who has earned his wealth, position and power the old-fashioned way, by marrying it. He seeks to remind us that he is the regular, white American; and his opponent is The Other.

It's just another, more subtle way of playing the race card.

You've heard the legal aphorism "grandfathered," meaning that some new rule will apply to new situations but people who started out under the old rule can continue under the old rule?
The origin of that name for that legal practice is the Jim Crow South. One way they kept black people from voting without using a more overt racial bias was to qualify voters on the basis of whether their grandparents voted.

If your grandfather voted, you could vote. If your grandfather was a slave, well, sorry ... nothing against you.