No. Saints Row 2 (PS3/360/PC) has no romance, dating, or partner system.
Character customization is extensive (body, voice, walk, clothing), and NPCs react to your respect/fear levels, but interactions are limited to:
Romance mechanics were never part of the Saints Row series until Saints Row IV (superficial "romance" dialogue options with crew) and the 2022 Reboot (relationship arcs with Lieutenants).