I don't fully understand what 'a one' refers to. The next line in the story explains that they were married for a long time. Does 'a one' refer to a married couple? "There once lived a king and a queen as many a married couple has been." Does 'a one' refer to being either a king or queen? "There once lived a king and a queen as many either a king or queen has been." I agree with Merlyn that the author is probably saying that the king and queen are similar to other kings and queens that have lived before. This is an old Scottish story from the 1800's. It's possible that the grammar is not accurate or is different than what we would expect to see today.

There have been may kings and queens such as these that have lived before

Thank you for your comment.

As to the grammar of this, "one" can be used to refer to both at the same time, as in "many a one queen and many a one king"