... you have some considerable misunderstandings about the general nature of networks there. What you wish to do might well still be possible though via various means.
Yeah generally speaking anything on the internet will ignore local anything in the designated ranges of local IP addresses, indeed will refuse to send it. Though for the record you can set whatever IP addresses you want locally and have a local network access it -- local IP ranges are just the agreed upon thing outside your world. You can call your bedroom the white house and use an internal post system to send a letter to it if you want, same idea here.
Anyway two approaches people tend to use here
1) Port forwarding.
2) Virtual private network aka VPN.
There may be a third effort in the future if/when IPv6 takes off but I will skip that for now. I will also be skipping the more interesting aspects of tunnelling.
1) Your router has an IP address the world knows it as that your ISP provides. Sometimes for a fee they will give you a static IP as opposed to once every 24 hours to a week giving you one from their pool of them (dynamic IP) which can make life a bit easier for some things, though most will counter that with a dynamic DNS (ddns) service (services that your router/PC/device on it say this is me every few minutes and have the ddns service update their corresponding web address for rather than needing to know your router's external IP at that moment.
Your router will also have a firewall/port forwarding that allows certain ports to speak to the outside as normally it blocks everything coming in unless it is expecting it (see NAT/network address translation if you are bored and want to know about that). You need to figure out what ports the PS4 uses to allow remote play and set the router to forward the relevant ports from said PS4* out to the world.
Does mean anybody can try to connect to your PS4 though (fine for a quake server or website you run out of your house, less good for your private PS4) and I don't know if the PS4 has passworded remote play (or if any password setup it has is any use). There are limited options for protection here (unlocking ports being the major one) and most not much use.
Sometimes ISPs will block certain ports as well (ports 80 and 25, being normal www web server and email respectively, being the most common but there are others and game servers are one of them)
*depending upon the router your PS4 might benefit from having an internal static IP so you can say "forward these external ports to this internal device", though other routers will allow you to use the mac address or device name when port forwarding.
2) VPN is not just a fairly pointless service for most people that your chosen video creator will shill from time to time but a concept with a long history in corporate and remote control.
The VPN will be established such that the network (or maybe just device) and home network (or maybe just selected device) are basically on a local network despite not being on the same physical network. Companies use them all the time to allow people to access databases they don't want out in the world, printers, file shares, login details, internal email and whatever else. I don't know what you might have to play with as far as joining a VPN with a Switch, though there might be some homebrew. Alternatively some might VPN with a phone/tablet/laptop and use the phone/tablet/laptop to set up a hotspot that the Switch connects to.
Even if you do either of those you will run slap bang into the major problem though
https://www.gamasutra.com/view/feature/3374/the_internet_sucks_or_what_i_.php?print=1
It is old but it is not like physics has changed in the subsequent years. If the PS4 streaming options are not built to handle the awfulness of the internet (your little home router is far far far nicer as an environment).
The PS4 might even have blocked the attempt -- the xbox 360, following many doing it for the original xbox with stuff like xlink kai, blocked all local network traffic above a certain ping (hacked versions could dodge it, and you only needed one in the pair to be hacked) to stop people from trying to do LAN play games over longer distances.
If it is just the other side of town then you might be able to get away with it.
More generally I assume we have all seen the general failure of streaming games thus far -- all the lag and whatnot. This is physics at its heart that is the problem rather than coding. Though a server farm in every major city or section thereof would alleviate that for at least those with such toys.
You mentioned teamviewer. It is a class of program known as virtual network computing aka VNC, though colloquially remote control** is what most will call it. Indeed this whole PS4 streaming play is arguably a version of this. This could be an option too, if you want to remote control a PC on your network that is in turn remote controlling the PS4 then you could both dodge the lag as it would be locally controlled and if the connection cut out you just dial back in. Could get tricky as you would be forwarding lots of video data and trying to get controllers working over the internet is a nightmare most of the time (though options exist to turn controller into key presses and mouse movements and then back again the other side).
**technically you can remote control a computer with SSH, indeed I am not scared about a hacker controlling my mouse but actually a hacker having such access and me not seeing it happen despite sitting at the machine while it is going down -- imagine what I can do if I have a command line access that is invisible to you) but let's not go there for now. SSH is good stuff though and worth knowing about if you are learning about this sort of thing.
