Some generic dongles operate at 2.4GHz
Kinda, sorta.
First, while Bluetooth (BT) is a wireless technology that uses radio frequencies (RF), it is a different technology than what is commonly referred to as "RF" devices. They are not compatible, or interchangeable.
Typical "RF" devices found in the home use frequencies from 900MHz up to 3.2GHz. There are other frequencies but those are the most common for things like RF keyboards and mice, TV remote controls, cordless phones, cordless headsets, and more.
Some may indeed use 2.4GHz too (like BT), but those typically are already "paired" in the firmware at the factory for the device they came with. This is to ensure if a customer buys 5 keyboards at once that each are on a unique frequency and channel so they don't interfere with each other. Can you imagine the chaos if one person in the room typed a letter and that same letter appeared on everyone's screen? Or they moved their mouse and everyone's pointer shot across the screen?
While interference is possible, it is not very common because the 2.4GHz band is divided up into different "channels". Bluetooth has 79 different channels, each with 1MHz bandwidth. When pairing, the devices look for unused channels. One of the major differences (and advantage) with Bluetooth is the user, with most BT devices, can manually pair the device with the receiver - just about any BT receiver. Not so with RF.
All these wireless devices "pair" using unique identifiers too to ensure they can work in the same proximity with other devices without interfering. "RF" has about 65,000 different unique IDs to use.
Technically, considering there are 100s of millions of computers out there (and even more wireless devices), it is possible two devices will use the exact same frequency, channel and ID. This is rarely a problem because the effective range of these devices is so small - 100 feet would be a stretch. But if it happens with BT, the user can typically re-pair and come up on a different channel.
Automobile keyless entry systems (key fobs) work in similar ways, most commonly using either 315MHz or 433MHz frequency bands. Can you imagine stepping out of Walmart and pressing your key fob button to find your car and EVERY car in the lot honked?
I note the OP said this was a "cheap" keyboard and mouse set. Factory set "RF" pairing is less expensive to produce than BT so odds are it is RF and not BT. Therefore, it cannot be re-paired with a different dongle.