SQL Server statistics
Including multiple columns in statistics adds a new dimension that cannot be estimated otherwise: correlation.


Query Predicate Contains Multiple Correlated Columns


  When a query predicate contains multiple columns that have cross-column relationships and dependencies, statistics on the multiple columns might improve the query plan. Statistics on multiple columns contain cross-column correlation statistics, called densities, that are not available in single-column statistics. Densities can improve cardinality estimates when query results depend on data relationships among multiple columns.

Optimizing Queries That Access Correlated datetime Columns
Exploring Column Correlation and Cardinality Estimates



  , if a column is replicated in more than one of these statistics and the first statistic is updated, how does this update affect the other statistics


All statistic need to be updated to be accurate.

Whether all stats recommended by the DTA are necessary, that is a different topic.