This is a general solution that assigns dense levels on partitions to rows. It uses user variables:

CREATE TABLE person (
    id INT NOT NULL PRIMARY KEY,
    firstname VARCHAR(10),
    gender VARCHAR(1),
    age INT
);

INSERT INTO person (id, firstname, gender, age) VALUES
(1,  'Adams',  'M', 33),
(2,  'Matt',   'M', 31),
(3,  'Grace',  'F', 25),
(4,  'Harry',  'M', 20),
(5,  'Scott',  'M', 30),
(6,  'Sarah',  'F', 30),
(7,  'Tony',   'M', 30),
(8,  'Lucy',   'F', 27),
(9,  'Zoe',    'F', 30),
(10, 'Megan',  'F', 26),
(11, 'Emily',  'F', 20),
(12, 'Peter',  'M', 20),
(13, 'John',   'M', 21),
(14, 'Kate',   'F', 35),
(15, 'James',  'M', 32),
(16, 'Cole',   'M', 25),
(17, 'Dennis', 'M', 27),
(18, 'Smith',  'M', 35),
(19, 'Zack',   'M', 35),
(20, 'Jill',   'F', 25);

SELECT person.*, @rank := CASE
    WHEN @partval = gender AND @rankval = age THEN @rank
    WHEN @partval = gender AND (@rankval := age) IS NOT NULL THEN @rank + 1
    WHEN (@partval := gender) IS NOT NULL AND (@rankval := age) IS NOT NULL THEN 1
END AS rnk
FROM person, (SELECT @rank := NULL, @partval := NULL, @rankval := NULL) AS x
ORDER BY gender, age;

Please note that variable assignment is within a CASE expression. This (theoretically) solves the evaluation order problem. IS NOT NULL was added to handle data type conversion and short-circuiting issues.

PS: This can be easily converted to row numbers on the partition by removing all conditions that check for ties.

| id | firstname | gender | age | rank |
|----|-----------|--------|-----|------|
| 11 | Emily     | F      | 20  | 1    |
| 20 | Jill      | F      | 25  | 2    |
| 3  | Grace     | F      | 25  | 2    |
| 10 | Megan     | F      | 26  | 3    |
| 8  | Lucy      | F      | 27  | 4    |
| 6  | Sarah     | F      | 30  | 5    |
| 9  | Zoe       | F      | 30  | 5    |
| 14 | Kate      | F      | 35  | 6    |
| 4  | Harry     | M      | 20  | 1    |
| 12 | Peter     | M      | 20  | 1    |
| 13 | John      | M      | 21  | 2    |
| 16 | Cole      | M      | 25  | 3    |
| 17 | Dennis    | M      | 27  | 4    |
| 7  | Tony      | M      | 30  | 5    |
| 5  | Scott     | M      | 30  | 5    |
| 2  | Matt      | M      | 31  | 6    |
| 15 | James     | M      | 32  | 7    |
| 1  | Adams     | M      | 33  | 8    |
| 18 | Smith     | M      | 35  | 9    |
| 19 | Zack      | M      | 35  | 9    |

db demofiddle

One option is to use a ranking variable like this:

SELECT    first_name,
          age,
          gender,
          @curRank := @curRank + 1 AS rank
FROM      person p, (SELECT @curRank := 0) r
ORDER BY  age;

(SELECT @curRank := 0) section allows variable initialization without a separate SET command.

Test case:

CREATE TABLE person (id int, first_name varchar(20), age int, gender char(1));

INSERT INTO person VALUES (1, 'Bob', 25, 'M');
INSERT INTO person VALUES (2, 'Jane', 20, 'F');
INSERT INTO person VALUES (3, 'Jack', 30, 'M');
INSERT INTO person VALUES (4, 'Bill', 32, 'M');
INSERT INTO person VALUES (5, 'Nick', 22, 'M');
INSERT INTO person VALUES (6, 'Kathy', 18, 'F');
INSERT INTO person VALUES (7, 'Steve', 36, 'M');
INSERT INTO person VALUES (8, 'Anne', 25, 'F');

result:

+------------+------+--------+------+
| first_name | age  | gender | rank |
+------------+------+--------+------+
| Kathy      |   18 | F      |    1 |
| Jane       |   20 | F      |    2 |
| Nick       |   22 | M      |    3 |
| Bob        |   25 | M      |    4 |
| Anne       |   25 | F      |    5 |
| Jack       |   30 | M      |    6 |
| Bill       |   32 | M      |    7 |
| Steve      |   36 | M      |    8 |
+------------+------+--------+------+
8 rows in set (0.02 sec)