Translation:My wife likes to buy clothes, but I hate that.
Hah. Good question. I think most probably the latter, but I wouldn't exclude the first interpretation.
Wouldn't it be correct to say ".....but I can't stand it" instead of "...but I hate that."?
Close in meaning, but I think not enough - after all, hatred is a very, very strong emotion! I would say "nie mogę tego znieść" for "I can't stand it", if we were shopping right now, or "nie znoszę tego" more generally.
Why is "ja" necessary? Doesn't "nienawidzę" show that it is 1st person singular?
It does, but the subject suddenly changes in the middle of the sentence, so you actually do need to put it explicitly here.