Need monitor for MacBook Air 13" 2013

MacBook Air (13-inch, Mid 2013) - Technical Specifications - Apple Support

"Simultaneously supports full native resolution on the built-in display and up to 2560 by 1600 pixels on an external display, both at millions of colors"

That Mac supports one external display. It has a Thunderbolt 1 port with a Mini DisplayPort connector which can double as a Mini DisplayPort.

I'm not sure why you would need software for the HP and Samsung monitors, if you were only trying to hook up a single monitor at a time, using a Mini DisplayPort connection. Most monitors (though not all) have hardware menu controls on the monitors themselves.

"Need monitor for MacBook Air 13" 2013: Have tried two 24-inch monitors and they can be connected via hardware with the MacBook Air 13" running macOS Mojave 10.14.6, but software for the monitor is not available to run on the aforementioned Mac.. The two monitors were made by HP and Samsung."

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Getting a Monitor to Work:

Get an Adapter:

Hardware seems to be the method of what to troubleshoot here. Here is how to set up your Mac with an Apple 30" Cinema Display: Configuring a USB-C Only Mac with a 30-Inch Apple Cinema Display - User Tip. There is one adapter that equals out the wattage to make this all work: Uptab USB-C to Mini DisplayPort Adapter 4K@60Hz - Silver

A 13" 2013 MacBook Air does not have USB-C ports.

"A 13" 2013 MacBook Air does not have USB-C ports."

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Correct. I'm suggesting it is a hardware error, and an example as to how to fix this for a Mac.