Aladdin's wish for the grail is to obtain the grail. In this form Aladdin is on his quest to retrieve the lamp and whenever he hears "wishgranter," he just assumes it is the lamp even if it's not. But at the same time, there really isn't much difference between two supposedly omnipotent wish-granting vessels so does it really matter?
Aladdin is a rather weak Heroic Spirit considering his feats are more about being lucky than combat or skill. He has been shown to be quite clever and he did win an actual war without his lamp so he's not completely useless.
When summoned Aladdin is a restless and mischievous soul who is slightly weighed down with his need to regain his lamp (the grail). The obsession grows and grows each day until it utterly consumes him, after all this Aladdin is the embodiment of a reversal of fortune. Becoming more and more ruthless and desperate, the only ending for him is one of ruin. Which is ironic because of how he overcame such a predicament in his legend. Well, maybe if he has an understanding Master who balances him out.
Design-wise, having something be able to grant all your wishes was way too OP. So I tied that with his theme. As the war drags on for longer and longer, he gets weaker and weaker, but also stronger in a different sense. Basically, he's trapped by a specific moment in his legend his only escape from that hell is to win or die. He totally remembers everything about his legend, but maybe that just makes him all the more desperate, since he knows he's regained the
before.
Yeah, you probably shouldn't summon Aladdin as an Assassin. Well, he's a fun guy at the beginning though.
Btw
Ifrita =