Remote Screen Sharing Stuck on Connecting

To access a network client on a local network from a remote location over the Internet, at least two things will be required:

1. A static WAN-side IP address provided by your ISP. This is typical for business-grade Internet service. Most, if not all, consumer-grade Internet service are provided with dynamic WAN-side IP addresses. To get around this, you would employ Dynamic DNS (DDNS) so that you would use a domain name, instead of an IP address, to gain access to the router from a remote location.
2. Screen sharing is a network service. As such it needs to communicate through a number of ports. To access the local device, you would need to configure your router for port mapping/forwarding to be able to receive screen sharing traffic from the Internet through those ports on your router's firewall.

As a brody has already mentioned, you can make life a lot easier by using commercial service, like TeamViewer, to do all the "dirty" work for you. Otherwise, you will need to get your "hands dirty" to understand how to do both of those items I mentioned yourself.

It definitely could be your router.

Screen Sharing supports connectivity when the IP address is mapped on the correct ports on the router, and you have the password to access the Mac. The ports aren't necessary when both Macs are on the same subnet or have a non-10.x.x.x or non-192.x.x.x IP address. 169.x.x.x indicates no internet access is present. The ports ar defined on this support article:

Screen sharing type options on Mac - Apple Support

Kaseya, Anydesk, Teamviewer are three third party utilities that offer a variety of cross platform screen sharing options to internet connected computers. While not as fast as IP over Thunderbolt or IP over Firewire, or LAN based sharing, they will at least give you the opportunity to do file sharing, screen sharing, and control the other computer from another computer using a passcode and unique registration code.

Yep, port forwarding is the term. I would contact your ISP's tier 2 support for information on setting up port forwarding on your router. They may have suggestions.