Short answer, "no"; medium answer, "there's a typo in the class description". Long answer:
Eldrich Knight's spell progression (along with every spell-level-progressing prestige class I can think of) says:
From 2nd level on, when a new eldritch knight level is gained, the character gains new spells per day as if she had also gained a level in whatever arcane spellcasting class she belonged to before she added the prestige class. She does not, however, gain any other benefit a character of that class would have gained. This essentially means that she adds the level of eldritch knight to the level of whatever other arcane spellcasting class the character has, then determines spells per day and caster level accordingly.
So, a Wizard 4/Fighter 4/Eldrich Knight 4 would have the spells slots of a 7th level Wizard (remember: the first Eldrich Knight level doesn't increase spells), but would have only one bonus Wizard feat, and her familiar would have an Int of 7, +2 natural armor, and would not yet be able to speak with his master. Additionally, she wouldn't have received 6 "free" spells for her spellbook.
Taking a level of, say, Archmage would increase her Wizard caster level, thus giving her spell slots of an 8th level Wizard, but her familiar would remain unchanged.
If you don't take all 10 levels of Anima Mage, you never get Vestige Casting.
Vestige Metamagic (Su): At 4th level, you can use a bound vestige to augment your spells.... Vestige metamagic is usable once per day at 5th level, two times per day at 7th level, and three times per day at 9th level. (Tome of Magic, p50-51)
Similarly, Wiz 4/Anima Mage 4/Eldrich Knight 2 (and whatever else for the prerequisites) would have the spell slots and save DCs of a 9th level Wizard, but would still only be a 4th level Anima Mage and, thus, unable to use Vestige Metamagic.
... however ...
The Vestige Metamagic description almost certainly has a typo in it somewhere; either it should start with "At 5th level..." (which would match the later description and the table), or the text should say that it "is usable once per day at 4th level...".
Another way of looking at it
Neither Anima Mage nor Eldrich Knight increase your "Wizard level", but they both increase your ... "Wizard Caster Level", for lack of a better term, which Wizard also does.
If a character had both Wizard and Sorcerer levels, they would need to choose whether to increase their "Wizard Caster Level" or "Sorcerer Caster Level" when they took a level of Anima Mage (but, note that they could decide independently each time they took a level). This is the great power of Mystic Theurge: it increases both an Arcane (Sorcerer/Wizard/Bard/...) and a Divine (Druid/Cleric/Paladin/...) Caster Level each level; however, the trade-off is that it doesn't get the Wizard/Druid theurge any additional feats, a better familiar, increased Wild Shape ability, or an enhanced animal companion.