Here's something I copied. Seems to make sense. I haven't really played any games with any good Udyrs, so I honestly have issues recognizing what stance he is using.
Suggested Role - Tank (Initiator) / (Jungle) Ganker
Right off the bat you should notice that Udyr's abilities are well-suited for primarily a Tank role. With damage mitigation from Turtle Stance and Monkey's Agility and the ability to initiate on opponents with Bear Stance, Udyr is a solid choice when a team lacks a tank. He has decent AoE damage output from Phoenix Stance which allows him to be a threat especially in team-fights. In any Udyr build either Phoenix or Tiger Stance must be chosen as the primary source of damage. If you were planning on more of a Carry-type build, Tiger Stance could work but for a Tank build it is not synergistic whatsoever.
His only weakness as a pure Tank is that he has no large AoE to initiate with and has to rely solely on Bear Stance instead to single out opponents. But what Udyr lacks in that department he makes up in his ability to jungle and Gank extremely well. Although he lacks the true burst of a traditional Ganker, Udyr makes up for this with the extreme burst movement speed and lockdown of Bear Stance combined primarily with the Red Lizard buff which is nearly impossible to escape. Also because he is tankier than most Gankers, he can dive towers and bowl through enemy allies with much more ease. All of this allows him to be much more aggressive than most Gankers.
If Udyr can manage a strong early game typically via controlling the jungle and ganking effectively, he can definitely become an almost unstoppable force that can tank entire teams for the entirety of a team-fight while dealing a deceiving amount of damage through Phoenix Stance. The spammy nature of his abilities allows him to be a consistent presence that can easily outlast and eventually overwhelm most other champions. In this regard, you could consider him somewhat a Semi-Carry but I'd rather not delve into that debate.
In the mid / late-game, Udyr's role is the same as any other Tank. Udyr wants to favorably initiate on the enemy team hopefully catching a valuable target off-guard so that he can continue on to stun them, disrupt them, and perhaps eventually land a handful of kills. Udyr presents a tough choice for the enemy team. They can choose to try to ignore you which will allow you to spam mainly Phoenix and Bear Stance to deal potent damage while also disrupting the enemy or they can try to focus you which will be fended off by Turtle and Bear Stance spam in order to reduce their damage output and get out of the fray when it gets too dangerous.
