Is this poor sportsmanship? Chessmanship?

If someone accidentally makes a move (in this case I accidentally moved my queen once space forward into the attack of both a pawn and a knight; the attack was obviously not on purpose), should the other person accept a takeback?

There is no obligation to accept takebacks. I almost never accept the request. These games are next to meaningless, so unless it was massively interesting, I just say, "be more careful next time."

Im not a fan of takebacks, you made a mistake. It might be an honest mistake but it happened. Same as if you missed a mate in one and lost, you shouldnt want to undo it but prevent it from happening in the future. As Delpire said, be more careful.

I usually accept takebacks if it was a very obvious mouseslip.
Otherwise I do not.

Basically it's like in a football game when the goalkeeper drops the ball in his own goal post. Is it possible to take back the ball? Yes, of course, but the goal still remains.

I think the sportsman-like behavior on part of the player who made the mistake is to resign the game and ask for a rematch, and not to ask for a takeback.

Yea, it is better to resign. Anyway, accepting such kind of requests may lead to an infinite loop off takebacks, which is unacceptable.

Takebacks are for mouseslips only, in unrated games, between friends.
Otherwise ever regret can be taken back and it makes the game meaningless.

But on the other hand, asking for a trackback during a casual game seems to be fair enough.

I no longer accept take backs from any opponent. There is no unwritten or written rule that says these requests should be honored. In my fair share of giving into the requests the player usually played a completely different piece, which signifies the original move is a blunder and they don't want to lose the piece.

Bottom line, if you don't wan't to be declined on your take backs, then put move confirmation on. Usually you see what you missed as soon as you set the piece, the extra step of confirming this is the move you want to make can save your skin often enough.