There’s a girl who lives in a house or has just moved to a house, and in the house lives a cat. She goes up a staircase into a room with many doors in it, and through one of the doors, the cat brings her to a world full of snow. I believe there are other doors with other places through them but I can’t remember. This was a picture book I believe and likely was written in the 90’s or even early 2000’s.
Almost certainly ‘Zoom Away’ by Tim Wynne-Jones and Eric Beddows (1992). The middle of a trilogy!
Not a picture book, but “House of Many Ways” by Diana Wynne Jones has a similar premise and some illustrations, and was publisehd in the first decade of the 2000s. It’s an older children’s/young adult fantasy novel about a girl called Charmian who goes to house-sit for her absent great-uncle, who is a magician. His house is like that.