Robins are an all year round bird in the UK:
with its bright red breast it is familar throughout the year and especially at Christmas! Males and females look identical, and young birds have no red breast and are spotted with golden brown.
So this proves, for those of a keen eye, that there's inherently nothing special about the Robin Male vs Female, only the juveniles have a different colouring, no changes throughout the seasons like some mammals, nothing. They're pretty dapper in their red chests all year.
Although robin folklore existed for many years, the robin became far
more widely associated with Christmas after it was depicted on
Christmas cards in 19th century England; a tradition that has been
retained to this day.
Why are robins on Christmas cards? It’s all inspired by the robins who
used to deliver the Christmas cards in 19th century Britain.
In the 1800s, British postmen wore bright red uniforms. They wore red in honor of the crown since red is considered both a Royal color and an important color on the English flag. (Incidentally, this may also be one of the reasons why British post boxes were eventually standardized to be red). The postmen in their red-breasted coats resembled the much-loved British bird, the robin red-breast, earning Victorian postmen the nick-name: Robins.
Ok so this is a nice piece of information I did not know - Posties have always had a nickname of sorts, and it could be a valid reason for why they are on Christmas cards etc around that time of year, however there are other speculations too.
Legend has it that the robin's redbreast gives it a direct link to Christianity.
One fable suggests that when the baby Jesus was in his manger in the stable, the fire which had been lit to keep him warm started to blaze up very strongly. A brown robin, noticing that Mary had been distracted by the inn-keeper’s wife, placed himself between the fire and the face of baby Jesus. The robin fluffed out its feathers to protect the baby, but in so-doing its breast was scorched by the fire. This redness was then passed onto future generations of robins.
Further reading: Saga website on Christianity and Robins
I think as well one good idea about why they are on postcards is when we used to get a huge blanket of snow Mr Robin Redbreast would stand out against the bland colours of winter with his bright red feathers, the contrast looked good and someone painted it and... here we are!