Guys, thanks to an advise of one user from another forum, I fixed the issue:
So, the problem is that there was no (automatically assigned) iCloud account - "Me" with my phone and email. I have my own full name n the contacts , but it is not named just "Me". So, I set/add new iCloud contact and named it just "Me" and add my phone and AppleID email. And voila - The iMac and iPhone start "seeing" each other and the AirDrop is working now as a charm.. Obviously this is a iOs bug (missing the Me contact )