This is fairly obscure, but (like any good riddle) obvious when you have the right information to see the answer.
Have a look at The Form of Solemnization of Matrimony from the Church of England's Book of Common Prayer: essentially, the stuff that a priest would say during an Anglican wedding ceremony.
I publish the Banns of Marriage between M. of - and N. of -. [...]
I N. take thee N. to my wedded wife, to have and to hold from this day forward, for better for worse, for richer for poorer, in sickness and in health, to love and to cherish, till death us do part, according to God's holy ordinance; and thereto I plight thee my troth.
I N. take thee N. to my wedded husband, to have and to hold from this day forward, for better for worse, for richer for poorer, in sickness and in health, to love, cherish, and to obey, till death us do part, according to God's holy ordinance; and thereto I give thee my troth.
In this pretty official document, "M." and "N." are used as placeholders to represent the names that the priest should read out at the beginning of the ceremony. Later on, when the couple are taking their vows, it's only "N." that's used, but other versions of the Book of Common Prayer (such as this one from c. 1800-1820 - thanks @GarethRees) consistently use "N." for the bride and "M." for the groom. Note the appearance of the phrase "for better, for worse", so that in the versions which consistently differentiate between "N." and "M.", we literally have the line "I N. take thee M. [...] for better for worse", almost exactly the same as in your quoted passage. This confirms that we've solved the riddle correctly and found what's being referred to.
So yes, it is "an abbreviation of the names of a couple getting married", as you guessed. But not a specific couple: rather, "N." and "M." are essentially the "John and Jane Doe" of Church of England documentation.
Thanks to this site for pointing me to the correct answer. It also contains further information about the usage or non-usage of "M." in different editions:
In the earlier versions of the prayerbook, this "N." occurred wherever either the man's name or the woman's name was to be spoken as part of the wedding ceremony. In most later editions of the prayerbook, in order to prevent any possible confusion as to where the man's name was to be spoken, and where the woman's name, two different letters have been used -- "M." was introduced to mark places where the man's name should be said, while "N." was left to mark places where the woman's name should be said. The differentiation was probably done in this way merely because the man's name is usually said before the woman's name in the ceremony, and "M." comes before "N." in the alphabet; as far as I'm aware, there is no deeper significance to the particular choice of letters for the man's and woman's names, and the letter "M." doesn't seem to abbreviate anything (in the way that "N." can be said to abbreviate nomen). (One ingenious suggestion, that "M." and "N." were intended to stand for Latin maritus "husband" and nupta "bride", must remain rather doubtful.)