I just purchased a "generic" female esata - female sata cable, I didn't know anything about "plain" or not cables, honestly I saw only "eared" esata plugs on the web until now. Can you please show me which esata form cable should I get? An amazon or aliexpress link would be ok so I won't be wrong.

Let me understand better why mine doesn't work, especially why the fantec box's led remains switched off. As far as you said my cable (with "ears") takes energy from the mobo so I guess the reason is there's "something" within the fantec that avoid it to get power from its power supply while connected via this "eared" cable, isn't it? Moreover I guess if I didn't connect its power supply it'd not work anyway because the energy taken from the "ears" would be not enough, right?
Just a "punk" solution: can't I just isolate or cut/interrupt these 2 wires? I bought this cable only for this goal but it's useless as it is.
Consider that I've already tried to connect it with both cables at the same time (usb 3 and esata-sata II). The fantec's led switched on and it was recognized connected via usb therefore it was as there were no sata cable. Can this be useful to get informations about how this "thing" works?

Finally a last note about these attempts, maybe this'll be useful maybe not.. Every time after I use this new esata-sata cable and replace the hdd inside the PC case or I use the usb cable to connect the fantec box, after rebooting the PC the CPU mobo led switches on and the booting freezes. This doesn't happen if I use only the usb cable. When this happens maybe I can restart just pushing the PC case reset button maybe it doesn't work and I've to switch off the PC. This [switching on - cpu led - switching off] can go on for 2-3-4 times without a rule (I guess). Most of times I solve just switching off once and waiting 10 minutes, then the PC boots properly.
Can this behavior be useful to understand better the relationship between my mobo and this esata cable?