Do noroviruses show the same seasonality as flu?

Yes they do. Noroviruses used to be called winter vomiting disease, because they mainly occurred in the winter. Influenza is so called because of "Influenza del freddo" - the influence of the cold in Italian,The only time you get noroviruses significantly in the summer months is when you get a new strain from an RNA mutation. More usually you get infections between October and March.