In thirteenth century French a 'cote' was a small shelter. This meaning came into English and the suffix-age was added to mean 'the entire property attached to a cote'.
To know what type of gardens were attached to such places during the Middle Ages, we must turn to what is known about medieval gardens: not much. But it is known that most of the land was used for vegetables and if flowers were grown it was only if they a medicinal or culinary use. This applied even to the rose and the lily.
It was not until the late nineteenth century, with the gilded vision of Arts and Crafts romantics, that a Cottage Garden became a thing to yearn for. Since then, there has been no looking back. All the world loves a cottage and few remember the generations of grinding toil. The truth of the matter is that the type of dwelling illustrated in the paintings belonged to prosperous members of the middle class. The rubble huts with earthen floors and leaking roofs inhabited by the peasantry have all gone.
The icon of musical rebellion, Bob Dylan, confessed in his autobiography (Chronicles, 2004) that he wanted nothing more than what sounded like a cottage with roses growing round the door
Kent cottage garden (Sissinghurst)