The robin is a symbol of Christmas in Britain, though I'm not sure why. It lives here year round, but it's not the only bird to do so. There are explanations. One is a connection with early postmen, who wore red coats — therefore robin red-breasts on cards. It's not completely convincing. Another is a legend about the robin keeping the fire going in the stable to keep the baby Jesus warm and its breast catching fire. A sweet tale, but the association of the robin with Christmas seems to come in Victorian times.
Perhaps it's simply because it's a friendly (to humans, at least) little bird with a cheerful colouring in that red breast. Except that its breast isn't red. It is, if you look, distinctly orange!
Why then, so famously red? I've heard that the word "orange" came late to English and that "red" was used for orange, but according to the OED Robin Redbreast crops up in the mid 16th century at about the same time that "orange" comes into use, both as a fruit and a colour.
So it's a mystery! But the robin does have a very benign reputation, as these quotes show.