What does "The type T must be a reference type in order to use it as parameter" mean?

If you look at the definition of Db<TEntity>:

public class DbSet<TEntity> : DbQuery<TEntity>, IDbSet<TEntity>, IQueryable<TEntity>, IEnumerable<TEntity>, IQueryable, IEnumerable, IInternalSetAdapter 
where TEntity : class

Because it has a type constraint that the generic type must be a class then you must initialize it with a type that also matches this condition:

public class GenericRecordController<T> : Controller where T : class
{ ... }

They apparently have a constraint on the generic type.

All you need to change is:

public class GenericRecordController<T> : Controller where T : class

This tells the compiler that only reference types may be supplied as a type for T.

You can do it on just a method as well:

        public bool HasKey<T>(T obj) where T : class
        {
            return _db.Entry<T>(obj).IsKeySet;
        }