Shido is Kotori's adopted older brother. Her deep love and care for her brother has been shown many times, but, however, she treats her brother lowly when she is in "commander mode", but clings to him heavily when in "little sister mode".
Kotori developed her "commander mode" persona when Shido gave her black ribbons as a birthday present, and made her promise to be strong when wearing them, since without them she proved to be quite a crybaby. However, when she is wearing her white ribbons or no ribbons at all, she returns to her persona as his younger sister, which proves to be Kotori's true self. A lot of Shido's peers actually find it strange how close they are. According to the Fraxinus' machines, her affection points for Shido neither rose nor fell during their date to seal her Spirit powers: because the said points were already maxed out the entire time.
Kotori can be considered a tsundere towards her older brother, often teasing him and even acting a little sadistic towards him while in commander mode. Despite her usually blunt and harsh attitude, however, Kotori's feelings for Shido run deep, and have been shown numerous times to go beyond the boundary of sibling love, to the point of being romantic. An example of this is shown in Encore 1 of the Light Novel (OVA 1/Episode11 of Date A Live II) when Yoshino told Kotori about a girl who got involved with Shido and was about to be with him forever, she immediately became enraged after hearing this, stating to herself that "no other woman will dare take away her brother" from her. Another is when she unintentionally told Mana during their argument in Volume 3 (Episode 7 of Date A Live) that "a real sister wouldn't even be able to marry him", which she immediately took off and changed the topic of before Shido and Mana realized what she said.
Kotori and Mana are considered rivals, mainly on who is a better little sister for Shido.
[To be Added...]
[To be Added...]