He's right that you don't need an actual inverter, but I don't understand why he got mad instead of explaining. I'll try to, even though you may or may not want to go with this solution.
Your battery/alternator will be supplying something like 12-14V DC. Your router needs 9V DC, which is just a bit below what the car supplies.
With an inverter you'll be converting those 12-14V DC to 110V AC and then back down to 9V DC which means quite a bit of power lost in such conversions. An (easy) alternative would be something like a buck converter, which would simply step down your car's voltage a bit to the 9V DC your router requires, bypassing those conversions. As a bonus, no need to worry about sine waves. You'd be using the buck converter instead of your router power adapter.
I'm not the best person to pick one, but based on a quick Amazon you have this; 3 of them for $14. With a led display because I assume you don't have a multi-meter. You'd still need a small screwdriver to adjust it until it said 9V, which is what your router requires.
That said, you'd need to be slightly familiar with electronics in order to both get a cigarette lighter plug to wire into the buck converter, as well as a jack that fits your router. If you have a friend that's comfortable with base electronics, you could just ask them.
The linked buck converter doesn't need soldering like the one in the video. You just screw the wire into place which is 100% easier. Def need the 3 pack. The first one the screws were kinda jank for me but the 2nd was perfect. Make sure to only wire in the red and black from the usb cable for power to the raspberry. Would also recommend setting the voltage to 5.2vs output on the buck converter. I had it set to 5.1 but klipper kept saying I had some undervoltage issues and that was resolved by changing it to 5.2.
I use a cheap adjustable buck converter from Amazon to provide 5V on one of my printers. It powers a bunch of RGB LEDs.
It seems reliable enough, in that it's been driving these LEDs 24x7 for over a year. I looked at the output on a scope once and I recall the noise being fairly insignificant.
Lots of people use these, or similar, to power Pi's.
Here's what I bought: https://www.amazon.com/gp/product/B07WQJ2GD6