I think I asked the wrong question. I think this is perfectly normal for an multi-HDD external USB enclosure or any storage the more HDD's there are. The question should have been something like if the enclosure can show the HDD's exactly, or at least similarly, as if they are internally connected. I don't think this is possible. Or is it?
Anyway, the built-in eSATA did not recognize the device at all. Documentation says if the eSATA in my computer does not support multi-HDD (or something, forgot the name), the enclosure may read only one HDD or none at all. Well it was the latter for me.