Here are three key relationships between Tai Lung and other imprtant characters in the plot:
The relationship between Tai Lung and Master Shifu is a complex one surrounding the two different masks Shifu wears through Tai Lung's maturation which he inevitably struggles to separate creating a monster out of the once seemless cub found afoot to the Jade Palace. The snow leopard Tai Lung first regards Shifu as his mentor in the arts and father figure, which helps him feel like he belongs. His deep-seated pride, however, transforms into bitterness and betrayal against Shifu when Tai Lung is not given the chance to become the Dragon Warrior. As Tai Lung strives to reconcile his expectations with Shifu's actions, their relationship depicts themes of betrayal, hatred, and broken identity. Ironically enough, it was Shifu's overboarding pride in Tai Lung as his "child" which invaded and imparied his foresight as a mentor forgetting to instill the phenom with the humility necessary for all warriors to strive and act appropriately in the face of adversity and rejection.
Although throughout the film, they have little interaction, it is ultimately Oogway who is tasked and able to succeed in restraining Tai Lung's rage in response to his judgement of Tai Lung. Oogway is seen as an almost infintely wise figure and thus determined Tai Lung's unworthiness in receiving the scroll. Tai Lung's subsequent outburst provides an ironic element to the movie as we find it only proves Oogway's point as he "saw darkness in his heart and refused".
Po exists as the main character of the trilogy, and Tai Lung as the main antagonist of the first, although upon deeper analysis their rivalry stems from misfortune. The different villains in the trilogy exist from different timelines:
Although Tai Lung may be interpreted as jealous of Po, and feels the urge to assert his dominance and strength over him, it is only a mere cause to his natural persona but more importantly in order to spite his former master Shifu as he sought to beat out Shifu's "new favorite", and the one who took the position he arrogantly thought was his: Dragon Warrior, heir to the Dragon Scroll.