Methods of cleaning
Current washing technologies are mostly solvent based. Most normal "soiling" of clothing is a mixture of oils and salts (Both of which are products of sweat), and sloughed skin cells, often bound by those same oils. When one finds clothing irritable from extended wear, it's usually due to the effects of accumulated sweat and skin cells, and the bacteria that eat them.
Normal washing uses a mix of water (as a solvent) and a surfactant to bind oil to the water for more complete cleaning. The agitation is mostly to move the water and surfactant mix through the garment and its fibers.
Dry Cleaning uses any of a number of chemical solvents to remove the various oils, salts, bacterial and skin cells.
It is possible to use abrasive processes to remove skin and oil accumulations; it won't generally remove them from between fibers, but "sand-washed" is a viable option for extending the wearability of a garment short term, at the expense of long term damage to the garment.
Air-washing is also possible; the use of high pressure and temperature air can remove large portions of the cells and bacterial colonies. It can remove some of the salts and oils, as well.
Evaporative cleaning can also be used; this can be used to remove oils, but won't remove particulates between the fibers.
Thermal degradation is used for certain forms of cleaning. That is to say, you heat it until the biologicals break down. It's not generally conducive to clothing, but in combination with other techniques, can be used for enhancement of other methods.
Why not use?
Water? It's present. In theory, astronauts could rinse garments with extant systemry. The thing is, water and other fluids are not easily handled in microgravity (as the wringing a washcloth video shows). It's certainly useable for limited applications. It's best used for removing salts; salt accumulation is primarily an artifact of sweat evaporation in a high heat environment, and is almost a non-issue in space. Plus, water is a limited quantity.
Surfactant? Most laundry surfactants are contact irritants. A spill in station would be a major hazard. Further, water used with them requires different purification from normal. Some specific ones could be used, but again, the environmental risks due to breathing in the material make it a risk, as does the increased risks to the water purification.
Dry Cleaning? Almost all the chemicals are highly toxic.
Air-washing? Noisy, and requires particulate capture aboard station. Also, it's not highly effective.
Evaporative? Slow, doesn't remove particulates (including bacteria and skin cells, as well as crystalline salt). Plus, anything removed now has to be scrubbed from the atmosphere. But at least, given enough time, the bacteria starve.
Thermal Degradation is a heating issue - both for the garment and for the habitat. Heat management is a major part of spacecraft design.
If you're going to anyway...
A combined pressurized steam to recapture system, possibly followed by vacuum drying, might be a reasonable system.
Use of pressurized steam gets maximum use of both thermal and air washing, plus removes some salts and oils from even inside the fiber bundles of the fabric, for a minimum amount of water. It's also a reasonably small system; a tractor feed unit could be created that washes a shirt at a time with a tractor feed. It also is of limited hazard to crew.
Adding vacuum drying on a warm surface would maximize volitalization of remaining liquids.