Store the name in the $this/$self object, not in the global $_name. $_name was a single variable shared between multiple objects and thus you were always overwriting the value whenever you called myNameIs.

package Interesting;

sub new {
    my ( $class ) = @_;
    return bless {}, $class;
}
 
sub myNameIs{
    my ( $self, $name ) = @_;
    defined $name
        or die "Don't tell me that I don't have a name.";

    # Store the name into the object
    return $self->{name} = $name;
}
 
sub whoAmI{
    my ( $self ) = @_;

    # Retrieve the name from the object.
    return $self->{name};
}
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Separately, I would like to urge you to stop using the syntax $one = new Interesting in favor of $one = Interesting->new. There are some tricky bugs related to the first syntax and you can avoid that entire class of errors by just not using that syntax. If there were a new() function in the same package as the code which had new Interesting, then instead of calling Interesting::new( 'Interesting' ) it would call WhateverPackage::new( 'Interesting' ). In OO code, if you had another class with a method named new(), then you might find that the incorrect new() method was being called.