Clothier's closet, is a 2nd level Conjuration(creation) spell, whose description says:
...The conjured clothing is normal in all respects and does not radiate magic. Even after the duration elapses and the rod and hangers [the clothes are conjured on] disappear, the clothes remain.
So they persist indefinitely as (20% real) normal clothes. The spell says that they do not radiate magic, so they don't (that's a bit of 'specific trumps general').
The SRD version of shadow conjuration regarding objects says:
You use material from the Plane of Shadow to shape quasi-real illusions of one or more ...objects[.] Shadow conjuration can mimic any sorcerer or wizard ...conjuration (creation) spell of 3rd level or lower.
Shadow conjurations are actually one-fifth (20%) as strong as the real things, though creatures who believe the shadow conjurations to be real are affected by them at full strength.
Any creature that interacts with the conjured object, force, or creature can make a Will save to recognize its true nature.
A creature that succeeds on its save sees the shadow conjurations as transparent images superimposed on vague, shadowy forms.
...any effect created by shadow conjuration allows spell resistance, even if the spell it is simulating does not. Shadow objects or substances have normal effects except against those who disbelieve them...Against disbelievers, they are 20% likely to work.
Objects automatically succeed on their Will saves against this spell.
So, clothier's closet, being a 2nd level Conjuration(Creation) spell, obviously qualifies for being simulated by the shadow conjuration spell. The clothes have 20% of normal hit points, and as DM, I'd rule hardness as well (which for most cloth should be 0 anyway). A creature trying to tear them off a character or otherwise damage them would have an easier time of it as they have 80% fewer HP.
If you're wearing the clothes, and someone looks too hard at your outfit or touches it, they get a save to know it's not real. So anyone making the save can see they're not real.
Creatures with spell resistance, when it's successful, don't perceive the rod and hangars at all anymore if they touch them, but the clothes, persisting past the spell's duration are no longer subject to spell resistance. The pertinent rules regarding spell resistance here are:
Magic actually has to be working for spell resistance to apply. Spells that have instantaneous durations but lasting results aren’t subject to spell resistance unless the resistant creature is exposed to the spell the instant it is cast.
Against an ongoing spell that has already been cast, a failed check against spell resistance allows the resistant creature to ignore any effect the spell might have. The magic continues to affect others normally.
As an object, the warrior's sword will always cut these illusory clothes easier than normal clothes, since it gets a save as an attended object, which it automatically succeeds at, and thus it ignores 80% of any hardness the clothes might have, and they have 80% lower HP.
Crai's comment regarding clothes
that have game effects recalled that some clothes are useful as well as decorative:
Artisan’s Outfit has pockets which will only hold an item 20% of the time. (Attended objects get saves, objects automatically succeed at the will save to disbelieve, and I'm not sure that the attender can willfully forgo that save...)
Cleric’s Vestments have no listed effects.
Cold Weather Outfit the bonus to saving throws could be lessened from +5 to +1, as weather is an object. However, it's unattended, so perhaps just leave it.
Courtier’s Outfit does not include the 50 gp of jewelry, so you still look like an out-of-place commoner without it.
Entertainer’s Outfit is what this spell might be 'perfect' for. Though 25% of folks looking at the outfit instead of the entertainer will then focus again on the entertainer (upon making their saving throw).
Explorer’s Outfit: also has pockets that might be less than useful 80% of the time.
Monk’s Outfit has folds (pockets) too, and the sash-ropes are 80% weaker and also completely fail to affect attended objects 80% of the time and fail versus creatures ~20% of the time.
Noble’s Outfit, like the courtier's outfit, also has no signet ring or jewelry; without them you'd still not fit among other nobles.
Scholar’s and Traveler’s Outfits are actually pretty simple, being good clothes plus cap and hood.
Regarding the impact of free clothes on the economy, consider that shadow conjuration
is a 4th level spell, and its save DC is often 16.
So 25% (5 of 20) average folks examining or touching the clothes will know they're fake. Higher wisdom or higher level folks will have a better chance to see that they're false. Those with spell resistance effectively are unable to use them [SR]% of the time, unless they choose to forgo their SR.
Additionally, word of mouth will have everyone looking for false clothes once it's known that these false clothes exist, and that 25% can communicate their success to others, granting a +4 bonus to disbelieve for those around them.
So, if overused they'll eventually be hard to sell, and even without overuse, they're not terribly useful.