Correct me if I am wrong but I thought "benziyor" strictly means "he/she is looking like" in which case the translation would be "He/she is looking a lot like me", however this is marked as incorrect. Granted, it is more natural in English to say "he/she looks a lot like me" however it is not necessarily inappropriate to say "is looking" rather than "looks"; for example, a woman might be trying on different dresses and her friend says to a third person: "now she is looking like me". Interested in replies.
I understand phlippery's confusion and the explanation you make does not really explain the situation. How do we say "I love you" in Turkish? That's a present simple sentence. Yes, we say "Seni seviyorum" and not "Seni severim". In Turkish, we mostly use present continous structure for present simple expressions. A few more examples....
O kızı her gün görüyorum.: I see that girl everyday. / Biliyorum: I know. / Geceleri sesler duyuyorum: I ear sounds at night. / Seni anlıyorum: I understand you. / Özel bir şirkette çalışıyorum: I work at a private company. / Yazları hep aynı yere gidiyorum: I always go to the same place in summers. / Beş yıldır araba kullanıyorum: I drive for 5 years. / Katılmıyorum: I do not agree. / Et yemiyorum: I do not eat meat. / Çocuklara bayılıyorlar: They adore kids / Sizi yeterli bulmuyoruz: We don't find you sufficient. / Nerede yaşıyorsun?: Where do you live? / Sanmıyorum: I do not think (so).
I was trying to say that the verb "benziyor" does not strictly have a meaning in Present Continous Tense. This phenomenon exists not only in Turkish but also in many other languages, as well. For example; "The launch is taking place next month." (Future).
On the other hand, there are also a group of verbs, that cannot be articulated with -ing suffix. These verbs are called Non-continuous Verbs, and love is one of these stative verbs. The most of the example sentences you gave in the second paragraph falls into this category.
About this example, the verb "to look" falls into "Mixed Verbs" category. Depending on the context, it can or cannot be articulated with the suffix "-ing".
• He looks smart.
• He is looking at me.
In this example ,"looks like" seems to be only grammatically correct answer.