A quick search on Paizo's source reference document tells me that the Wild Shape class feature of the druid works just like the Transmutation spell Beast Shape.

Transmutation spells are described here and this is the relevant part:

When you cast a polymorph spell that changes you into a creature of the animal, dragon, elemental, magical beast, plant, or vermin type, all of your gear melds into your body. Items that provide constant bonuses and do not need to be activated continue to function while melded in this way (with the exception of armor and shield bonuses, which cease to function).

Unless some special feature like Wild lets him, your druid should not be gaining those bonus.
Wild armor is expensive (16000gp worth of enchanting) and I don't know how a 6th level character could buy it.

As @ZeDemonPyro makes me notice, that druid can't possibly turn into a dire tiger, because of the limitations in size and hit dice of the creatures a druid can wildshape into.

No, the effects of the spell ironwood aren't permanent nor does the spell permanency, as printed, allow ironwood to be made permanent (although a DM "may allow [it] to be made permanent" as an unlisted other spell, probably at a cost of 15,000 gp like other 6th-level spells (e.g. symbol of fear, symbol of persuasion)). That means you're absolutely correct in your assessment of the weirdness and nigh-uselessness of the spell. The only thing that I can figure is that...

The Game Makes Assumptions

Other than niche uses, such as a Magnificent Seven (or Three Amigos) scenario in which fortifications must be made to defend the town against an attack that's coming within the next few days (but not today because a 6th-level spell is meaningful and better spent), the spell ironwood is just a bad spell.

The problem is the game's assumptions. First, the game assumes the best armor normally available for a druid is hide armor. Second, the game assumes the druid must be completely self-sufficient, making everything--including the best armor--by himself out in the wild.

These aren't unreasonable assumptions, by the way, but they are flawed for most games. If the first is untrue ironwood's a dumb spell, and if the second is untrue ironwood's a useless spell.

The Druid Can Wear Better Armor

Even if no other options are available, the special material dragonhide can be used to make dragonhide plate, which functions as masterwork full plate, so a druid needn't use the spells ironwood and wood shape to make a suit of full plate unless he's lost his dragonhide plate, which is difficult in combat (although not impossible--there are ways to strip him of it) or the culmination of a convoluted series of events out of combat.

A druid might feel better having crafted his own armor, even if he must use his highest level spell every 11 days to renew the armor's effect, and I guess that's important. The only real reason I can see for a druid to do so is if the campaign world excludes or makes exceedingly rare special materials like dragonhide, forcing the druid who wants to wear another kind of armor besides hide or leather to find another source for that better armor. And, dude, that source is, like, totally nature.

The Druid Can Buy Better Armor

The Player's Handbook for Pathfinder's antecedent Dungeons and Dragons 3.5 didn't include dragonhide and other special materials; those were in a separate Dungeon Master's Guide. Thus it was possible for druids in that game to never even know of the existence of dragonhide armor or any of the myriad other kinds of armor better than hide they could wear. Pathfinder, by moving special materials into the chapter Equipment alongside daggers, 10-ft. poles, and spell component pouches, eliminated any mystique surrounding such items and eliminated--possibly--a major reason for the spell ironwood.

"Why's this spell even here?"

That's a really good question, and I can't even speculate very well. The spell ironwood doesn't appear in the Player's Handbook for Advanced Dungeons and Dragons nor Advanced Dungeons and Dragons, Second Edition. It does appear in the Wizard's Spell Compendium for Advanced Dungeons and Dragons, Second Edition, but its description calls wizards wu jen and notes that the spell is "[c]ommon in oriental [sic] settings; very rare elsewhere" (469), which leads me to believe that it originally appeared (the Wizard's Spell Compendium being a collection of previously published spells) in either Oriental Adventures for Advanced Dungeons and Dragons or an Advanced Dungeons and Dragons, Second Edition source covering the same material, but I can't speak to that.

The ironwood spell in the Wizard's Spell Compendium is a 5th-level alteration spell with a touch range, a 1-round casting time, a permanent duration, and an Area of Effect entry of 1 cubic foot per level. It grants that affected wood "the tensile strength... of the finest steel" (468). If it's used on a living plant, the plant continues to grow metal leaves and bark. If used on a plant creature, the creature's paralyzed if it fails a saving throw. So in older editions it's mainly a toy--a useful tool for DMs to explain why a wizard has steel trees in his garden and other weird decorations (there were a lot of spells like that in older editions) and as a sit-back-and-watch item for the DM to put on a scroll that the PCs will find, just to see what sort of shenanigans they'll try with the spell. DM fiat played a much larger role in spell adjudication in the game's older versions.

But, as can be seen, this older version of the spell ironwood--with its permanent duration and affecting a cubic foot of wood per level--would be a lot more fun and useful than the version Dungeons and Dragons, 3rd Edition--and, eventually, Pathfinder--saddled us with. I've no idea Jonathan Tweet's intent when he chose this previously obscure spell for inclusion in the Dungeons and Dragons, 3rd Edition core rules yet rewrote it to be largely useless.