I believe the author of the translation could not imagine a situation where 'já hledám svoji ženu' would not be something I am currently doing thus 'am looking'.
As for 'svůj, svá' vs. "můj, tvůj, jeho, etc.'. This is a tough one for native speakers, so do not feel too bad.
In general, if the object belongs to the subject of the sentence, 'svůj' should be used. John is looking for his wife. In English you cannot be sure, if the wife is John's or if John is looking for Ben;s wife. In Czech if John is looking for John's wife, it would be 'SVOU'. If John is looking for Ben's wife, it would be "JEHO'.
This rule has a bunch of exception. I am putting on my coat. If for some reason I need to point out that the coat is really mine, because Jane has the same one, I can say "oblékám si MŮJ kabát'. If I do not need to point it out, I can simply say that 'oblékám si SVŮJ kabát'. So you see why native speakers struggle at times as well. Welcome to the club!
I would argue that there is no real ambiguity of who's wife John is looking for in the sentence "John is looking for his wife"; if the sentence is alone, then John can only be looking for his own wife as no other person has been introduced and, therefore, cannot be the "owner" of the wife. If another person has been introduced, then the preceding sentence(s) will clarify the ownership. "You see that person over there? John is looking for his wife." - here it wouldn't make sense if John was looking for his own wife.
A somewhat gray area could be found in the following conversation: "Have you seen Ben?", "No, but John is looking for his wife, maybe she knows where Ben is". Here it could actually be possible that John is looking for his own wife; it could be common knowledge that John's wife was among the last persons to see or talk to Ben (and, thus, would be a logical person to ask), it could be common knowledge that John's wife often hangs out with Ben (so finding her would likely mean finding Ben), or maybe John's wife is known to be so good at finding people that she would be the one everyone want to ask for help. However, unless there is a clear, preferably outspoken, reason why John's wife would be the person to find Ben, the sentence can only mean that John is looking for Ben's wife. Without the extra knowledge that John's wife is the best person to find Ben, it would be highly illogical for John to look for his own wife when Ben's wife is far more likely to know where her husband is.
As for the real reason I came to this discussion: I, too, would argue that "I am looking for" and "I look for" is interchangeable here and that "I look for my wife" should be accepted. I wouldn't say "I look for my wife" unless it is something I'm currently doing.
We learned hledat as "to look for" something. I see no reason why "to search for" is not also a good way to translate hledat. At least one online dictionary provides it as a translation:
search 1. sb sth prohledat koho/co 2. for sb/sth hledat koho/co, pátrat po kom/čem
I reported it, and would appreciate any feedback.