If I understand correctly: (a) the "Live" version of Zorin 16 on the USB connected properly; (b) the connection persisted throughout the installation process; but (c) the installed version of Zorin 16 on the computer does not connect.
The most likely causes are:
(1) the Ethernet driver that was installed was the correct driver but did not install completely/correctly; or
(2) the Ethernet driver that was installed was not the correct driver; or
(3) the connection protocols between the computer and the router aren't negotiating correctly.
You are working with @sugaratias on the connection protocol issue.
Here is a suggestion that might be worth trying to narrow down a possible driver issue:
You reported the results of sudo dmesg | grep eth0 from the computer after installation:
[ 0.640933] r8169 0000:03:00.0 eth0: RTL8168g/8111g, a0:d3:c1:48:ed:67, XID 4c0, IRQ 31
[ 0.640936] r8169 0000:03:00.0 eth0: jumbo features [frames: 9194 bytes, tx checksumming: ko]
[ 0.641733] r8169 0000:03:00.0 eno1: renamed from eth0
Try booting your computer from the installation USB (that is, boot into a "Live" "Try Zorin OS") session and see if your computer connects to the Internet, as it did during the initial installation process.
If it does connect in the "Live" session, run sudo dmesg | grep eth0 while in the "Live" session and see if the results are different than the results you got running the installed version. If the results are different (e.g. show something other than "RTL8168g/8111g"), that might suggest a path to a driver that will work in the installed version.
Realtek drivers are a constant source of problems with Linux and the problem might take some serious effort to track down. I realize that is cold comfort, but once people with more experience than I have get the right information, a fix might be found.