Plugging two guitars into a guitar amplifier that is designed for one guitar at a time, can be done - but it's not recommended for reasons suggested in other answers.
You can find many guitar amps billed as "two channel" amps, but many of these are still intended to amplify only one guitar at a time. They might have a "clean" channel and a "dirty" channel and are intended for one guitar to be able to play through either one or the other. But they only have one set of controls so plugging a guitar into each channel is probably not a good solution for what you are trying to accomplish.
However, there are amplifiers available that have two independent channels - each with its own separate preamp and each with its own set of individual controls for the intended purpose of amplifying two separate instruments. Most of these true two channel/two preamp combos are billed as "acoustic" amps but are fully capable of amplifying any electric guitar. They just won't have the "dirty" distortion/overdrive effect that an electric guitar amp might have, but you can use a distortion pedal to get that effect if desired.
If you choose this solution, be sure the amp has two instrument inputs. Many of the "acoustic" amps have an XLR input for one of the channels so you can use it for guitar and vocal. But many of these also have an instrument input available to use on the channel with the XLR (use one or the other). You also want to be sure each channel has its own dedicated volume and tone controls.
Using a two channel amp with dedicated controls for each of two guitars would allow you and your student to both plug in to the same amp and then adjust the tone and volume independently for each guitar.
Below are some pictures of a few examples of the type amp I am talking about.
Another option that might even be more space efficient if you don't mind spending a little more - would be a line array portable PA system such as the Fishman SoloAmp SA 220 (pictured below) which has two channels with dedicated controls. Either of the two channels on the Fishman SA 220 can be used for an instrument or a microphone. Something like this could also be used if you wanted to set up a public recital for an audience to showcase one of your students.