Assume Supernatural Ability
A general feat (Savage Species p.30) available to be taken by any creature with the requisite Wisdom Score, and an "ability to assume a new form magically," Assume Supernatural Ability is contentious in its interpretation. This is partly because form altering magic or abilities have been fraught with difficulty, and partly because the feat is not descriptive, and gives no examples for clarity.
The Problem with Polymorph
Because Savage Species was published almost four months prior to the completion and release of the 3.5 Revision of Dungeons & Dragons in 2003, and had been in development throughout most of the time of the revision, it has intermediate rules information between the 3rd and 3.5 versions of the game.
Many creatures in 3.0 had polymorph self as a spell-like ability (Sp), while others had different, usually supernatural (Su) abilities, such as Polymorph, Change Shape, Alternate Form, or Alter Self, each of which often mimicked polymorph self, but also alter self or shapechange. With the 3.5 Revision, the polymorph self spell became the polymorph spell, and almost every instance of it, alter self, and shapechange was removed from creatures' lists of abilities. Instead they often became the precursors of the current Change Shape and Alternate Form abilities. Even then, there were multiple changes to polymorph, Change Shape, and Alternate Form until the designers eventually appeared to give up and make the 'Polymorph' sub-school, which they revised at least once more until producing the 4th edition.
For example, in 3.0, Doppelgangers had alter self as a (Su) ability at will, Avolakia (from Monster Manual 2) had (Sp) polymorph self (humanoid form only) at-will, Succubi had the same (Sp) as Avolakia, Grimalkin (MM2, p122) had (Su) Polymorph (as polymorph, into Medium or smaller animals, beasts (revised to animals, usually) and vermin), Imps had (Su) Polymorph (as polymorph, 1-2 forms, Medium or smaller), Quasits had (Su) Alternate Form (also as polymorph, 1-2 forms, Med. or smaller), and the Phasm had (Su) Alternate Form (as shapechange, Diminutive-Large).
Each of these creatures got either a form of Change Shape or Alternate Form, except the Phasm, which got its Alternate Form switched from the more powerful shapechange to polymorph. This seems to indicate to me that nearly all form altering abilities are 'similar' to polymorph.
Interactions between 3rd and 3.5 form altering abilities are, therefore, to put it in technical terms, an awful mess. This highlights why some players will say that most form altering magic that lets a creature appear to be another to meet the 'polymorph selfspell or a similar effect' metric of the Assume Supernatural Ability feat, and others adhere to only converting the words polymorph self to polymorph.
Vague Wording is Vague
Another reason that the feat's use is contested is that the benefit is nebulously worded, stating:
"You learn to use a single supernatural ability of another kind of creature while assuming its form through a polymorph self spell or a similar effect..."
The first sticking point is the '...a single supernatural ability of another kind of creature...' clause. The ways that this can be read are usually:
'choose one (Su) ability of a creature whose form you can assume, when you assume its form, you may use that ability'
'choose one type of (Su) ability that creatures have, when you assume the form of a creature that has an ability of the same name, you may use that ability'
'when you take the form of a creature, you may use any one (Su) ability that kind of creature normally has'.
The second sticking point is the '...while assuming its form...' clause. The taking of form is not strongly defined, with players debating how weighted the word 'form' is within the rules. Even the supposedly 'weak' shape altering magic of Eberron's Changeling, Minor Change Shape, is assumed to allow the changeling to take the form of a humanoid, as evidenced by the Racial Emulation feat (Races of Eberron, pp.110-111)
"...When you use your minor change shape ability to assume the form of a humanoid creature..."
Thus, the answer to the question of whether a Doppelganger can qualify for and use Assume Supernatural Ability is very difficult to definitively answer, but I personally would say 'yes'. Ultimately, the players, including the DM, of course, must decide whether to allow it.
In hopes of balance, I'd probably just add in a clause that only if your HD or Character Level are equal to or greater than that of the creature you're appearing as are you able to assume that creature's form, exactly as with form altering spells in 3.5 rules.
Finally, an Answer!
Regarding the last part of the Benefits section of the feat:
"...Using this alien ability is disorienting. You take a -2 penalty on all attack rolls, saving throws, skill checks, and ability checks. Additionally, in a stressful or demanding situation (such as combat), you must succeed on a Will save (DC 19) or be unable to use the ability."
This is a holdover from the 3.0 edition of polymorph other, which polymorph self inherits from. It has a very similar clause, which states (PHB 3.0, pp.236-237):
"...The new form can be very disorienting. Any time the polymorphed creature is in a stressful or demanding situation (such as combat), the creature must succeed at a Will save (DC 19) or suffer a -2 penalty on all attack rolls, saves, skill checks, and ability checks until the situation passes..."
The feat looks to actually state that you're 'disoriented' (with the same penalties as being 'shaken', by the way...) the whole time you're using the creature's ability. Thus if it's an attack, during the attack action, you take the penalty. If it's a constant (Su) defense or ability, you're always disoriented. Additionally, you must make the save or either lose, or be unable to use, the ability in a clinch.
That's why Improved Assume Supernatural Ability is in the same book. It has a higher Wisdom prerequisite, but states (Savage Species, pp.35-36):
"...This feat functions like Assume Supernatural Ability, except that you do not take the -2 penalty on all attack rolls, saves, skill checks, and ability checks when using it..."