It's besides the point if the Borg use polaron beams or not.
It's enough if their weapons had some properties similar to the polaron beams
And those properties are...?
if the shields were 'specifically designed to protect against that one thing'
No, just configured to protect against polaron beams. It's possible to reconfigure shields with a few keystrokes to defend better against different types of energies, once you know the nature of the weapon being used against you. This is exactly what was tried on the Odyssey by rotating shield harmonics, and is also exactly what is done to shields when trying to neutralize Borg weaponry (with varrying degrees of success).
That neither the Odyssey nor Defiant were able to overcome Dominion weapons suggests they have nothing in common with the Borg. Best evidence for this comes from the fact that even a Galaxy class starship is capable of partially deflecting Borg tractor beams and pulse weapons for a limited time, where Dominion weapons blast right through no matter what you do.
IOW: the only way Jem'hadar weapons are in ANY WAY similar to Borg weapons is if the Borg suddenly start using Jem'hadar weapons for some reason.
It's a well known fact SF modifies their existing tech to incorporate new things.
Which is exactly why Starfleet shields became effective LATER, after capturing and analyzing a Jem'hadar ship. They certainly weren't in The Search, before those modifications had occurred.
Of course, the major thing to consider is that the reason Jem'hadar weapons were so effective is probably because the Dominion went out of their way to infiltrate the Federation and steal their technological secrets so they could reconfigure their weapons to be more effective against them. This obviously would have included any experimental shield technology being bounced around the anti-Borg thinktanks (since those groups probably aren't classified given the Borg's disinclination for clandestine intelligence gathering).
The 'starboard power coupling' fails on virtually all SF ships at all times.
Sure. Just not
spontaneously. On Odyssey it was put out of action by enemy fire, and it is clearly intended to be the same case on Defiant.
Merely because there were some comparable instances to the Odyssey, doesn't imply the shields were useless to begin with ...
That's exactly what it IMPLIES. The question is whether or not that implication is supportable or not: is there any reason to assume they
weren't, other than speculation about Jem'hadar weapons possible similarity to Borg weapons (such similarity itself being speculation without any support of its own)?
I don't think so. In fact, we have Weyoun expressing surprise in "A Call To Arms" that DS9's shields were actually effective against Jem'hadar weapons, and he explicitly says that Starfleet shields HAD NEVER BEEN EFFECTIVE BEFORE THEN. This obviously includes their first encounter with the Defiant.
When I mentioned that weapons drain shields, I meant it in a way that the shields lose integrity when hit which is measured in % by SF (and virtually every other race).
Which is damage TO shields. Draining energy FROM them is something that is done only by the Borg. Furthermore, there is no reported power loss to the shields in either battle, and Defiant's shield failure was a result of damage to the ship itself, not the shields.
Also ... the Borg tractor beams drained the Enterprise shields fairly quickly in both of it's encounters ...
Except in the second encounter, where rotating shield modulation DID prevent the tractor beam from locking on for several seconds. Odyssey tried this exact same trick on the Jem'hadar, but to no avail. This directly implies two completely different systems at work here.
When shields are mentioned that they are failing...
It was not mentioned that they "are failing." T'rul simply says "Main power is offline! The shields have collapsed!" Seconds earlier, Odo mentioned as much to Kira: when main power goes down, shields go down; immediately after the failure of the shields, the Jem'hadar board the ship.
Prior to this, the shields seem to have no effect at all in blocking weapon fire, in fact Sisko gives the order to abandon ship almost immediately after the shields fail, when Defiant is already on fire from stem to stern.