'Through thick and thin' is an idiom which means in all obstacles and difficulties. The person who sings this song means to say that the girl will stand by him in good as well as bad times no matter what happens! 'Thick' is likely to imply the better times and 'thin' is not so good as the better times i.e. the difficult phase of life. Eg; I'm your wife and I promise to stick with you through thick and thin.

To my knowledge, it means " by all manners", "at whatever cost", at all account", "insistently". Hope, it helps.

Even when there are problems or difficulties: HE`S SUPPORTED THE TEAM FOR OVER TEN YEARS THROUGH THICK AND THIN. I hope that helps