Rank function in MySQL

One option is to use a ranking variable, such as the following:

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

The (SELECT @curRank := 0) part allows the variable initialization without requiring 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)

Here is a generic solution that assigns dense rank over partition 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;

Notice that the variable assignments are placed inside the CASE expression. This (in theory) takes care of order of evaluation issue. The IS NOT NULL is added to handle datatype conversion and short circuiting issues.

PS: It can easily be converted to row number over partition by by removing all conditions that check for tie.

| 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    |

Demo on db<>fiddle