In SQL, the WHERE clause does not take a list; it takes an expression. So replace the commas with 'AND's.

  UPDATE Geo_postaggregated_pop_count_SPLIT 
  SET    Total_person_count = t.Sum_count
  FROM   Geo_postaggregated_mortality_count_with_cause_catagory_SPLIT_
+SMR G,
         #tmp_G_P_M_CC_S_S t
  WHERE  G.Key_m                     = @Key
    AND  G.Time_instance             = t.Time_instance
    AND  G.Geo_class                 = t.Geo_class
    AND  G.Aggregated_area           = t.Aggregated_area
    AND  G.Super_classed_disease_cat = t.Super_classed_disease_cat
    AND  G.Cause                     = t.Cause
    AND  G.Constraint_ref            = t.Constraint_ref
    AND  G.Sex                       = t.Sex
    AND  G.Sex_code                  = t.Sex_code
    AND  G.Age_start                 = t.Age_start
    AND  G.Age_end                   = t.Age_end
    AND  G.Age_range                 = t.Age_range
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But whether or not this works, please take Mutant's advice.