Produce DISTINCT values in STRING_AGG

Here is one way to do it.

Since you want the distinct counts as well, it can be done simply by grouping the rows twice. The first GROUP BY will remove duplicates, the second GROUP BY will produce the final result.

WITH
Sitings
AS
(
    SELECT * FROM (VALUES 
    (1, 'Florida', 'Orlando', 'bird'),
    (2, 'Florida', 'Orlando', 'dog'),
    (3, 'Arizona', 'Phoenix', 'bird'),
    (4, 'Arizona', 'Phoenix', 'dog'),
    (5, 'Arizona', 'Phoenix', 'bird'),
    (6, 'Arizona', 'Phoenix', 'bird'),
    (7, 'Arizona', 'Phoenix', 'bird'),
    (8, 'Arizona', 'Flagstaff', 'dog')
    ) F (ID, State, City, Siting)
)
,CTE_Animals
AS
(
    SELECT
        State, City, Siting
    FROM Sitings
    GROUP BY State, City, Siting
)
SELECT
    State, City, COUNT(1) AS [# Of Sitings], STRING_AGG(Siting,',') AS Animals
FROM CTE_Animals
GROUP BY State, City
ORDER BY
    State
    ,City
;

Result

+---------+-----------+--------------+----------+
|  State  |   City    | # Of Sitings | Animals  |
+---------+-----------+--------------+----------+
| Arizona | Flagstaff |            1 | dog      |
| Arizona | Phoenix   |            2 | bird,dog |
| Florida | Orlando   |            2 | bird,dog |
+---------+-----------+--------------+----------+

---

If you are still getting an error message about exceeding 8000 characters, then cast the values to varchar(max) before STRING_AGG.

Something like

STRING_AGG(CAST(Siting AS varchar(max)),',') AS Animals

just use sub-query

WITH Sitings 
      AS
      (
        SELECT * FROM (VALUES 
          (1, 'Florida', 'Orlando', 'bird'),
          (2, 'Florida', 'Orlando', 'dog'),
          (3, 'Arizona', 'Phoenix', 'bird'),
          (4, 'Arizona', 'Phoenix', 'dog'),
          (5, 'Arizona', 'Phoenix', 'bird'),
          (6, 'Arizona', 'Phoenix', 'bird'),
          (7, 'Arizona', 'Phoenix', 'bird'),
          (8, 'Arizona', 'Flagstaff', 'dog')
        ) F (ID, State, City, Siting)
      ) 

    select State,City,count(*) as [# Of Types],STRING_AGG(Siting,',') AS Animals from 
    (
      SELECT State, City, Siting
    FROM Sitings 
    GROUP BY State, City,Siting
    ) as T  group by State,City

http://sqlfiddle.com/#!18/ba4b8/11

  State     City    # Of Types  Animals
Arizona     Flagstaff   1   dog
Florida     Orlando     2   bird,dog
Arizona     Phoenix     2   bird,dog

Here is one more way of doing it (sql fiddle):

  WITH Sitings 
  AS
  (
    SELECT * FROM (VALUES 
      (1, 'Florida', 'Orlando', 'bird'),
      (2, 'Florida', 'Orlando', 'dog'),
      (3, 'Arizona', 'Phoenix', 'bird'),
      (4, 'Arizona', 'Phoenix', 'dog'),
      (5, 'Arizona', 'Phoenix', 'bird'),
      (6, 'Arizona', 'Phoenix', 'bird'),
      (7, 'Arizona', 'Phoenix', 'bird'),
      (8, 'Arizona', 'Flagstaff', 'dog')
    ) F (ID, State, City, Siting)
  ) 

select State,City,count(*) as [# Of Sitings],(select string_agg(value,', ') from (select distinct value from string_split(string_agg(Siting, ','),',')) t) AS Animals
FROM Sitings 
GROUP BY State, City

You may easily convert the splitting and merging part into a reusable scalar valued function.

NOTE

This is NOT an optimal solution, if you group first and then do aggregate (like answers above) it is better. Also, it does not get # of Types, it gets # of Sitings instead. However, it becomes handy as a quick inline function.