Wizkids Rules!

When fielded, choose an opposing card, cancelling all previous choices. Your opponent may not purchase or field that character until Drax leaves the Field Zone.

If Drax leaves the Field Zone and returns that same turn, does he "remember" the choice he made?

No, he would not “remember” his target and must be fielded again to initiate his ability.

Drax’s "when fielded" ability is an applied effect (ability). Per the follow rule:

Applied effects are lost if a die is KO’d or rerolled to energy, even if that die returns to the Field Zone later in the same turn.

[Source, X-Men Forever Rule Book, Types of Game Effects section, p24]

By extension, this rule can be applied anytime a die leaves the Field Zone because of the portion of this rule that references a potential returning to the Field Zone would not enable the restating of the applied effect. This rule also covers other methods of leaving, such as being captured or removed by a specific ability - like Intimidate, and potentially returning to the Field Zone in the same turn.

Drax does not specifically state that his applied effect is regained once he returns to the Field Zone. Nor can it be implied the effect is regained because Drax specifically states the applied effect is lost when he leaves the Field Zone.

Therefore, if he is Intimidated, his 'prevent purchase/field' ability is lost. Drax's ability lacks any continuity language that would enable him to retain the prevention ability (i.e. being "while active").

Refer to Typhoid Mary - Red Rubber Boots or Shriek - Sonic Beam for examples of this continuity language.