Why it is implied that objects are equal if compareTo() returns 0?

Let's have a class Person. Person has a name and height.

Equals and hashCode() takes into account only name. Person is comparable (or we implement comparator for it, does not matter which one). Persons are compared by height.

It seems reasonable to expect a situation where two different persons can have same height, but e.g. TreeSet behaves like compareTo()==0 means equals, not merely same size.

To avoid this, comparison can secondarily look at something else if size is the same, but then it cannot be used to detect same sized different objects.

Example:

import java.util.Comparator;
import java.util.HashSet;
import java.util.Objects;
import java.util.Set;
import java.util.TreeSet;

public class Person implements Comparable<Person> {

private final String name;
private int height;

public Person(String name,
        int height) {
    this.name = name;
    this.height = height;
}

public int getHeight() {
    return height;
}

public void setHeight(int height) {
    this.height = height;
}

public String getName() {
    return name;
}

@Override
public int compareTo(Person o) {
    return Integer.compare(height, o.height);
}

public boolean equals(Object obj) {
    if (obj == null) {
        return false;
    }
    if (getClass() != obj.getClass()) {
        return false;
    }
    final Person other = (Person) obj;
    if (!Objects.equals(this.name, other.name)) {
        return false;
    }
    return true;
}

public int hashCode() {
    int hash = 5;
    hash = 13 * hash + Objects.hashCode(this.name);
    return hash;
}

public String toString() {
    return "Person{" + name + ", height = " + height + '}';
}

public static class PComparator1 implements Comparator<Person> {

    @Override
    public int compare(Person o1,
            Person o2) {
        return o1.compareTo(o2);
    }
}

public static class PComparator2 implements Comparator<Person> {

    @Override