Function of "with me" in this phrase
הָֽאִשָּׁה אֲשֶׁר נָתַתָּה עִמָּדִי follows the relative pronoun (אֲשֶׁר), which is modifying the woman (הָֽאִשָּׁה ); and the verb (נָתַתָּה ) is 2nd person, God is the subject. In comparison with other uses in Genesis of this form (עִמָּדִי ), about half of which follow the relative pronoun, it appears that the emphasis of the verb is usually on a superiors action "with me". The place of "with me" appears to indicate the passive involvement or inclusion of the "me" in the action of verb of the superior as in
Gen 29:19( אֲשֶׁר תַּעֲבֹד עִמָּדִי) cf. also "do with me steadfast love" (Gen 47:29). I.e., "with me" functions like an indirect object.
Looking at the verb and its meaning
Then, the question really becomes about the verb "thou gavest" in its use with God as the subject, the woman as the object of the verb and the man as the indirect object.
Having looked at all uses of
נָתַתָּה in the pentateuch, I find the closest use cases to be in verses indicative of harvest blessing given to the people from the LORD, and the land itself given to the people by the LORD. An example of this use is found in Deut 26:10:
וְעַתָּה הִנֵּה הֵבֵאתִי אֶת־רֵאשִׁית פְּרִי הָאֲדָמָה **אֲשֶׁר־נָתַתָּה לִּי** יְהוָה וְהִנַּחְתֹּו לִפְנֵי יְהוָה אֱלֹהֶיךָ וְהִֽשְׁתַּחֲוִיתָ לִפְנֵי יְהוָה אֱלֹהֶֽיךָ׃; and another very close use is in Gen. 15:3, "you have given me no descendent" (
לִי לֹא נָתַתָּה זָרַע). Gen 15:3 (just cited) is the only (other) place in the Pentateuch where a person is also indicated as gift.
It may also be reasoned that the source of the gift, "the giver" is really the benefactor to which everything belongs, when that "giver" is God.
I think it is interesting that most of the verb form's usage is in Exodus, in passages about the construction of the Tabernacle (ie, Ex 26:33, "you are to hang this curtain underneath the clasps" (
וְנָתַתָּה אֶת־הַפָּרֹכֶת תַּחַת הַקְּרָסִים). This usage is really not like the one you mention. But it does flesh out the range of use we've already indicated.
Conclusion: beautiful accompaniment (?)
In light of the above, I would view "with me" as being closer to the interpretation "accompanying me" than "belonging to me". The woman is the greatest gift to man, a blessing, she is a person.
I find it an interesting thought of how the curtains were "given" in their places; and would be inclined to take up such an image in my mind of the woman. The woman who blesses man with a beauty and completeness, without which he himself is merely what we call today, "a tool".
Unfortunately, Adam gave fallen man his chief example of failing to be a good husband; he casts blame on the wife whom God had given him. In that light, Adam probably wasn't thinking she was all that beautiful at the time. He would not have been wanting any more responsibility for his wife here, so that weighs against interpreting "belonging to me".
Rather, it is God who gave him this woman, and right now, he is just the victim in his own complaint, as 3:12 continues - "she gave it from the tree to me and I ate it!"