There are many options here, the question you have to ask yourself is how much work do you want to put into it?
A NAS is essentially a full computer that is self-contained in a nice little enclosure. You can either get one of the premade ones like the Synology one you mentioned, or build your own, or simply connect them to an old computer and share the hard drives with Windows SMB.
I will say this, the Synology one will be much easier to configure and manage. The DSM management software is very well made, just a point and a click to accomplish most tasks to get your stuff running.
As for the hard drive enclosure connected to your router, that depends on your router. Are you asking about an external enclosure to a router's USB port? If your router supports sharing external storage via its USB port, then yes you could. If it only supports DLNA then it won't work as a full NAS. You'll have to check with your router specs to know for sure.
I use an external hard drive dock to connect my HDDs to my computer.
From there, you could SMB or NFS share them on your network and devices could access them, but only when connected to a computer. Again, an old computer laying around would suffice if you really wanted a NAS. But the Synology would be the better option in my opinion.