Try adding the System.IO.Ports NuGet package:

<PackageReference Include="System.IO.Ports" Version="7.0.0" />

Also note that the fully qualified name is System.IO.Ports.SerialPort, not System.IO.SerialPort

For those coming to this post some years later...

Since .NET Core 2.0 it's possible to use the System.IO.Ports package.

An IO port is basically memory on the motherboard that you can write/read. The motherboard makes some memory available other than RAM. The CPU has a control bus which allows it to "tell" the motherboard that what it outputs on the data bus is to be written somewhere else than RAM. When you output to the VGA buffer, you write to video memory on the motherboard. The out/in instructions are used to write/read IO ports instead of writing to RAM. When you use out/in instructions, you instruct the CPU to set a certain line on its control bus to tell the motherboard to write/read a certain byte to an IO port instead of RAM.

Today, a lot of RAM memory is used for hardware mapping instead of IO ports. This is often called the PCI hole. It is memory mapped IO. So you will write to RAM and it will send the data to hardware like graphics memory. All of this is transparent to OS developers. You are simply using very abstract hardware interfaces which are either conventional (open source) or proprietary.

Serial ports in the meantime are simply ports which are serial in nature. A serial port is defined to be a port where data is transferred one bit at a time. USB is serial (universal serial bus). VGA is serial and others are too. These ports are not like IO ports. You can output to them indirectly using IO ports.

IO ports offer various hardware interfaces which allow to drive hardware. For example, if you have a VGA compatible screen and set text mode, the motherboard will make certain IO ports available and, when you write to these IO ports, video memory will vary depending on what you output to these ports. Eventually, the VGA screen will refresh when the video controller will output data written to video memory through the actual VGA port. I'm not totally aware of how all of this works since I'm not an electrical engineer and I never read about this stuff. To what I know, you can see the pins of the VGA port and what they do independently on wikipedia. VGA works with RGBHV. RGB stands for red, green and blue while HV stand for horizontal/vertical sync. As stated on wiki in the article on analog television:

Synchronizing pulses added to the video signal at the end of every scan line and video frame ensure that the sweep oscillators in the receiver remain locked in step with the transmitted signal so that the image can be reconstructed on the receiver screen. A sync separator circuit detects the sync voltage levels and sorts the pulses into horizontal and vertical sync.

The horizontal synchronization pulse (horizontal sync, or HSync), separates the scan lines. The horizontal sync signal is a single short pulse which indicates the start of every line. The rest of the scan line follows, with the signal ranging from 0.3 V (black) to 1 V (white), until the next horizontal or vertical synchronization pulse.

Memory in itself takes various forms in hardware. Video memory is often called VRAM (Video RAM) or the Frame Buffer as you can read in a Wikipedia article. So in itself video memory is an array of DRAM. DRAM today is one capacitor (which stores the data) and one mosfet transistor (which controls the flow of the data). So you have special wiring on the motherboard between the data bus of the processor and the VRAM. When you output data to video memory, you write to VRAM on the motherboard. Where you write and how just depends on the video mode you set up.

Most modern systems work with HDMI/Display port along with graphics card. These graphics card are other hardware interfaces which are often complex and they often cannot be known because the drivers for the cards are provided by the manufacturers. osdev.org has information on Intel HD Graphics which has a special interface to interact with. It can be used to gather info on the monitor and to determine what RAM address to use to write to the monitor.