As Swedish holidays and traditions may differ (more or less) from those in other countries, I will try to collect some general notes about these topics here, when I find that need of explanations arise in connection with postcards and photos.
NAME DAYS, GUSTAV VASA and 6th JUNE
Celebration of name days in Sweden began in the 17th century, at first among royalty and aristocracy, but then also among the general population. Each day in the calendar is associated with one or more names. A thorough modification of the list of names was made in 1901; but before as well as after, Gustaf (Gustav) was one of the names for 6 June (and still is).
The tradition of sending name day-cards may well have had a boost in connection with the 1901 alterations in the calendar. (Since my great-uncle's Gustaf's card collection starts the same year, I can't say if he received cards on his name day even before that.)
6th June has been celebrated nationally in Sweden since some time in the 19th century, in honour of the election of king Gustav Vasa on 6th June 1523, which is considered the foundation of "modern" Sweden. From 1916 the day was called Swedish Flag Day (Svenska flaggans dag). In 1983 it was elevated to official National Day; but not until 2005 was it also declared a public holiday.