The term "Scotch marriage" refers to a Scottish marriage "by declaration" before two witnesses, without benefit of a clergyman or government official. In the past such private marriages were legal, and in fact were very common; but they were dependent on the husband’s goodwill, to “do the right thing” and acknowledge the marriage openly.
In the case of runaway marriages by English couples travelling up to Scotland, the classic destination was the border village of Gretna Green, where a "blacksmith" presided over the ceremony and provided the witnesses. The marriage was registered in his book, rather than in the parish register.
So many legal issues were prone to surface following Scotch marriages that prudent parents would insist the young couple re-marry with a clergyman. The story of the Scotch marriage of Margaret Oliphant's great grandmother, Isabell Stewart, is told in Section 6 of The Family Tree of Margaret Oliphant. Perhaps harking back to Isabell's experience, Mrs Oliphant wrote several novels which include complications stemming from Scotch marriages.