Shift is problematic. I believe that it is not intended to give opportunity attacks with polearm gamble, but because of the way it was written this cannot be conclusively stated.
Forced movement and teleport do not give opportunity attacks. The wording on these two is clear and absolute and would require equally clear and specific language to be overridden.
"Enters" is a general term and is applied generally, just as "Leaves" is a general term when applied to the normal OA language. And "Shift", "teleport", "forced movement" are specific types of movement and the specific rules which govern them override the general rules for movement - in the case of Polearm Gamble and in the case of typical Opportunity Attacks.
If it was intended that Polearm Gamble did not follow the specific rules for teleportation and forced movement (and, imho, shifting) they would have added an additional specific rule to the effect of: This opportunity attack can be taken even if the movement was due to teleportation or forced movement (or, imho, shifting).
✦ No Opportunity Attacks: Your movement doesn’t provoke opportunity attacks.
✦ No Opportunity Attacks: Forced movement does not provoke opportunity attacks or other opportunity actions.
Shift, because it only mentions not triggering movement out of adjacent squares is less clear.
Ironically, however, by a strict and literal reading of the RAW - if the target shifts from a square that is adjacent to one of your allies to become adjacent to you, you do not get an OA.
SHIFT: MOVE ACTION
No Opportunity Attacks: If you shift out of a square adjacent to an enemy, you don’t provoke an opportunity attack.
Live by the strict and literal word, Die by the strict and literal word.
Or accept that strict literal reading of the rules can lead to interpretations that differ from what was intended when the rules were written and that, therefore, perhaps they intended shift to avoid movement provoked OAs whenever they occur and didn't hire a team of lawyers to make sure every rule was written with iron-clad language.