I would say no.
I read Enhance Potion as only affecting potions they drink. The jacket does not require a user to drink it. It's not drinking, it's more like, loading a syringe or syringe spear (unless there's a ruling that counts as drinking). The jacket absorbs and consumes the potion and then transfers benefits to the wearer.
Otherwise, it's not drinking, it doesn't provoke, it's an immediate action, a creature without a mouth can do it, and a creature that can't drink liquid or is harmed by touching liquid can do it safely.
it's tricky as you've put the potion in a magic item and then get the benefit on a round by round basis rather than drinking it as normal.
Poisoner's Gloves are similar in that you inject the potion into a target.
Based on that you'll need a generous Home Game ruling as a simple or pedantic reading says 'No'.
Absorbent fibers line this heavy canvas coat. Once per day as a standard action, the wearer can pour a potion onto this lining, suffusing it into the coat. If the potion has an instantaneous duration, the wearer can activate the coat as a swift action to consume the potion. If the potion’s duration is measured in rounds, the wearer can activate the coat as a swift action to gain the potion’s benefits for 1 round, repeating as desired each round until the potion’s entire duration has been used. These rounds do not need to be consecutive. The coat absorbs only potions with instantaneous or round-based durations, and only those which affect creatures (for example, it does not work on oils). Once the coat absorbs a potion, the potion cannot be retrieved from it, only used as described above. The absorbed potion loses its powers after 24 hours, even if unused.
Pouring the potion into the coat uses the potion and transfers its magic to the jacket. Once you pour the potion into the jacket the potion is gone. You cannot retrieve it and even if you don’t use the effect, it loses its power after 24 hours. Once you poor the potion into the lining it is no longer a potion.