Thank you. This is what I did and I still don't get the folders names like in Picasa. Does it mean it's not possible to import them?
You probably didn't "get" folders because Picasa doesn't make folders; it shows you the folders you have in the Mac filesystem. It's a browser. Like the Finder.
Photos, however, isn't. It imports files, and as noted above, by default it copies all photos on import into its own hidden folder structure. It then displays photos in albums, which do not have any necessary relation to folders in its library or in the Finder.
But if you turn off the "copy" button in Photos prefs, you can create a
referenced library, instead of a
managed one. Instead of copying, Photos stores a reference to a file somewhere, like in your Pictures folder. Sorta like using aliases.
So, if Picasa is showing a folder structure you already see in the Finder, and you want the pictures to remain there and be referenced by Photos, choose this option and turn "copy" off.
But then you are left with how Photos displays those referenced photos. It doesn't necessarily have to import them into an album that corresponds to a folder; you'd have to set it up to import folder "Mom's birthday" into say an album called "Mom's birthday." Subsequently you could add and subtract from the
album, but it wouldn't necessarily change where those images were in
folders in the Finder.
BTW, if you want to use iCloud Photo Library, you are gonna have to use the "copy" option; referencing won't work.
If this sounds like a mess, you might be better off using another product, like Picasa or Graphic Converter or Lyn (all browser photos), or AfterShot Pro, which can either browse or import.