For equals to work, you need to override hashCode also. Further equality check needs actual comparison of your objects.

Also, any related objects also has to implement hashCode and equals method. In this case Date.

Possible code

import java.util.Objects;

class Date {

    public int day;
    public int month;
    public int year;

    public Date(int d, int m, int y) {
        this.day = d;
        this.month = m;
        this.year = y;
    }

    @Override
    public boolean equals(Object o) {
        if (this == o) {
            return true;
        }
        if (o == null || getClass() != o.getClass()) {
            return false;
        }
        Date date = (Date) o;
        return day == date.day &&
            month == date.month &&
            year == date.year;
    }

    @Override
    public int hashCode() {
        return Objects.hash(day, month, year);
    }
}

public class Student {

    private final String firstName;
    private final String lastName;
    private Date birthDate;

    public Student(String firstName, String lastName, Date birthDate) {
        this.firstName = firstName;
        this.lastName = lastName;
        this.birthDate = birthDate;
    }

    public static void main(String[] args) {
        System.out.println(new Student("John", "Johnny", new Date(10, 10, 10))
            .equals(new Student("John", "Johnny", new Date(10, 10, 10))));
        System.out.println(new Student("John", "Johnny", new Date(11, 10, 10))
            .equals(new Student("John", "Johnny", new Date(10, 10, 10))));
    }

    @Override
    public boolean equals(Object o) {
        if (this == o) {
            return true;
        }
        if (o == null || getClass() != o.getClass()) {
            return false;
        }
        Student student = (Student) o;
        return Objects.equals(firstName, student.firstName) &&
            Objects.equals(lastName, student.lastName) &&
            Objects.equals(birthDate, student.birthDate);
    }

    @Override
    public int hashCode() {
        return Objects.hash(firstName, lastName, birthDate);
    }
}

Comparing objects with different fields sounds like bad polymorphic design, whether it's Java or any other OOP language:

• If your comparator needs to know the precise subtype of an object to do the comparison, you mess-up with the the open-closed principle, since for every new subclassing, you'd potentially need to modify the comparator to select the relevant fields.
• If your comparator needs uses reflexion to find on its own the relevant fields to compare, you indirectly mess up with the principle of encapsulation, since you create a hidden requirement that information to be compared must be in some predetermined field.

If you want to sort People properly in a clean polymorphic design:

• you need to rely either on a field, available for any kind of People, including Student, or
• you may call some function/transformation that provides a unique value (e.g. a string) that allows to sort any People. People and Student may then just use a different transformation that will be passed to the comparator; Or
• you only sort among homogeneous subtypes.

You need to Override equals and hashCode.
equals will compare the objects for equality according to the properties you need and hashCode is mandatory in order for your objects to be used correctly in Collections and Maps