Is Manny Machado The Best Young 3B Man Since Eddie Mathews?

Maybe. Here are all the seasons of 6+ WAR for 3B men through age 22 using the Baseball Reference Play Index:

Rk	Player	WAR	Year	Age
1	Dick Allen	8.8	1964	22
2	Eddie Mathews	8.3	1953	21
3	Eddie Mathews	7.8	1954	22
4	Jimmy Williams	6.9	1899	22
5	Manny Machado	6.5	2015	22
6	Manny Machado	6.4	2013	20
7	Hank Blalock	6.4	2003	22
8	Aurelio Rodriguez	6.4	1970	22
9	Freddie Lindstrom	6.3	1928	22

The only two guys to have two such seasons are Machado and Mathews. If the age is raised to 23, here is the list, Mathews has three and then Machado still has two. Only Dick Allen joins the group.

Rk	Player	WAR	Year	Age
1	Dick Allen	8.8	1964	22
2	Eddie Mathews	8.3	1953	21
3	Troy Glaus	7.8	2000	23
4	Eddie Mathews	7.8	1954	22
5	George Brett	7.5	1976	23
6	Eddie Mathews	7.3	1955	23
7	Evan Longoria	7.0	2009	23
8	Frankie Frisch	6.9	1921	23
9	Jimmy Williams	6.9	1899	22
10	Ron Santo	6.7	1963	23
11	Rogers Hornsby	6.7	1919	23
12	Scott Rolen	6.6	1998	23
13	Manny Machado	6.5	2015	22
14	Manny Machado	6.4	2013	20
15	Hank Blalock	6.4	2003	22
16	Aurelio Rodriguez	6.4	1970	22
17	Dick Allen	6.4	1965	23
18	Freddie Lindstrom	6.3	1928	22
19	Edgardo Alfonzo	6.2	1997	23
20	Gary Sheffield	6.2	1992	23
21	Eric Chavez	6.0	2001	23

Here is the top 10 in WAR through age 22:

Rk	Player	WAR	From	To	Age
1	Eddie Mathews	18.4	1952	1954	20-22
2	Manny Machado	16.9	2012	2015	19-22
3	Buddy Lewis	12.6	1935	1939	18-22
4	Freddie Lindstrom	12.5	1924	1928	18-22
5	Buddy Bell	9.6	1972	1974	20-22
6	Bill Dahlen	9.0	1891	1892	21-22
7	Bob Horner	8.9	1978	1980	20-22
8	Dick Allen	8.8	1963	1964	21-22
9	Denny Lyons	8.3	1885	1888	19-22
10	Adrian Beltre	8.2	1998	2001	19-22

If we raise the age to 23 we have

Rk	Player	WAR	From	To	Age
1	Eddie Mathews	25.7	1952	1955	20-23
2	George Davis	18.2	1890	1894	19-23
3	Manny Machado	16.9	2012	2015	19-22
4	Buddy Lewis	16.2	1935	1940	18-23
5	Dick Allen	15.2	1963	1965	21-23
6	Freddie Lindstrom	15.0	1924	1929	18-23
7	George Brett	14.5	1973	1976	20-23
8	Denny Lyons	13.8	1885	1889	19-23
9	Buddy Bell	12.5	1972	1975	20-23
10	Billy Nash	12.1	1884	1888	19-23