It's technically possible. The posixovl filesystem allows storing files on a FAT filesystem, with extra metadata stored in additional files to implement things that FAT doesn't provide: file names containing characters that FAT forbids or that are too long, additional metadata such as permissions and ownership, other file types such as symbolic links and devices, etc.
That doesn't mean that it's a good idea, though. It would be difficult to set up (I don't know of any distribution that sets it up for you) and slow.
Shortcuts could in theory be read as symbolic links, but this would have several downsides. Someone would need to write a filesystem driver that stores symbolic links as shortcuts. Windows might mess up symbolic links when it edits shortcuts (shortcuts are only very vaguely like symbolic links: symbolic links point to a file path, whereas Windows shortcuts track a file and Windows modifies the shortcut if the target file is moved). Linux would have no way to tell whether a file that looks like a shortcut is in fact intended to be a symbolic link or a regular file.
There used to be a way to install Linux on a disk image which is stored as a single file on a Windows system, called Wubi. It has been abandoned. It works, but it too has a number of downsides: lower performance, high risk of losing data if the system crashes, etc.
The normal way to install Linux is the best way: let the installer create a Linux partition. If you really, really don't want to create a Linux partition (for example because your corporate IT management forbids it), run Linux in a virtual machine. With Windows 10, you can run many Linux applications through the Windows Subsystem for Linux; you can get a whole Ubuntu userspace that way.