Not exactly but nearly so. Once it's setup, go to Network in Finder on the Mac. You should see it there with the name you assigned it. When you click on it you'll need to give it the user credentials you created when you set it up. You should be able to save these in the Mac Keychain so it won't ask again in the future. If you have multiple users you can set each one up with their own account on the Synology NAS and they will do the same when they want to use it but with their credentials.

Synology also provides desktop backup client apps for Windows, Mac, and Debian-based Linux distros, plus apps for Android & iOS so you can access the files or backup photos directly to it.

No, you're confusing a NAS with an external hard drive (or a hard drive enclosure, which turns an internal hard drive, or a Western Digital shucked hard drive, since their enclosures do not support Macs somehow, into an external drive).

An external hard drive (however you do it) plugs directly into your Mac. A NAS plugs directly into your router. Then it announces itself on the network. You connect to its address (I forget what the command is in Terminal — I have a Mac, too) and then set it up. You set a username/password, then you can connect to it remotely. In Finder you go to Go, then Connect a server, put in the IP address, then it'll ask for the username/password. Then just favourite it to the left, pin it, whatever. Finder will also remember the IP address. It says it'll remember it in the Keychain (which is now Apple Passwords, or maybe not) but my MacBook needed the password too, so I'm not sure what TF it was talking about.

Also, Firefox likes my NAS more than Safari does. Not sure what Safari's major malfunction is, but Firefox is my backup browser.

(Mine's a Netgear, not a Synology, so maybe a bit different.)

You want Localsend if you want to share specific files or you can selfhost a local filesharing service that is only accessible at home