Cardinals only live in the Americas, so while they'll have become a Christmas icon there, they're not well connected to it in Europe.

Europe has the European Robin however, affectionately called the Robin Redbreast in the UK, which has long been associated with Christmas and featured on some of the first Christmas cards.

The European Robin is mostly sedentary, so it doesn't migrate to warmer climates in the winter. When the trees are bare of leaves, it's quite easy to spot a Robin with it's colourful chest against the greys and whites of winter. This makes them quite an iconic image of winter.

It became a staple part of the modern Christmas tradition which started in the 1800s in the UK. It's certainly fair to say it's one part of why the colour red is associated with Christmas time.