I would try soap and water and a brush. The rubber itself looks to be in rough shape so if you care enough to clean them, I would just replace them. You can even upgrade the wheels while you're at it.
Keep the gray plastic. Remove black clip around the vertical wheel post in the gray plastic. Remove wheel and wheel post. Buy new wheel. Installl new wheel. It will be easier to find a new wheel once you have the old wheel out so you can take measurements. but it's likely something pretty standard, off-the-shelf. Wheels are something that companies buy, they rarely build them themselves. They typically come as a castoring assembly with wheel, axle, spindle, and attachment post in a variety of common sizes and with a dizzying variety of actual wheels.
When I clean the minerals out of a showerhead, I'll leave it in a bag of vinegar overnight or 8 hours over the day and agitate the bag every few hours(just a shake or two).
I don't really have it down to an exact science, I just use vinegar to clean non-precious metals and plastics because I know white vinegar dissolves dirt and grime well enough and dries without a smell.
Since the vinegar is going to get inside the wheel around the axles, I would probably dry them in white rice afterward after shaking off as much vinegar as possible, since you can do the same thing, just leave it in a bag of white rice overnight, the white rice draws out moisture really well.
Otherwise, I feel like some of the vinegar will still be trapped between the axle and plastic casing for a pretty long time and could poorly interact with any lubricant you spray in there.
It's likely hair causing issues. You need something harsh enough to dissolve it which won't dissplve the wheel itself and I'm not sure I can recommend anything. Nair or any other epilatory cream is likely to damage the rubber too. Vinegar will absolutely NOT dissolve the hair.
I hope some can help you disassemble the wheels properly. Good luck!
Disassembling the wheel: To do this, you'll need a screwdriver that fits the size of the screws holding the wheel in place. You should find them around the axle on the side where the wheel connects to the luggage. There might be more than one screw, so make sure you unscrew all of them. Once the screws are removed, gently pull out the wheel by applying pressure on both sides of the axle until it comes free. Be careful not to damage the wheel or its surrounding parts as you disconnect it.
Removing the inner components: After removing the outer casing (the part we see when the wheel is attached to the luggage), you'll have access to the inner workings of the wheel. The center hub should still be connected to the metal shaft, which is what makes the wheel rotate. This part may be stuck because of the accumulated dirt and grime. To separate them, apply some penetrating oil to the joint where the hub connects to the shaft. Wait for about 30 minutes before attempting to take apart these pieces.
Cleaning: Now that the inner components are separated, use a small brush and some warm soapy water to scrub away any remaining debris and grime. Rinse the parts with clean water, making sure no soap remains. Leave the wheel parts to dry fully before reassembling them.
Reassembly: When the parts are completely dry, reverse the process you followed to remove them, putting everything back together. Lubricate the shaft and hub connection with some light grease to prevent further gunk build-up. Make sure the nut on the shaft is tight enough but don't overdo it, as this could damage the threads.
Final assembly: Put the outer casing back onto the wheel, and then securely fasten the screws to attach it to your luggage again. Test the wheel by rolling the luggage around. It should now turn smoothly without any resistance from the built-up grime.
I would expect cheap luggage wheels to be press fit together. I expect the axle to not really be removable. Maybe it can be but I doubt it's designed to be. Chair casters are not serviceable.
My only evidence to this mode of thinking is "value engineering" in keeping the bill of materials down. Screws are laborious too.
Looks thick enough to scoop it off with a dull knife or spoon.
if you wanna use products, go from sfe to harmfull, depending on what the wheels are made of. You don't want to damage those.
I'd start with water. Soak em overnight in water, chances are it will come off easily after that. If not, you can try alcohol as mentioned in other comments. It looks thick, so it will need time to work in. Petrol or kerosine will work, but check compatibility with your wheels first.