It's probably not ozone.
Or if it is, they're lying about the shelf life of their product.
Laurel's answer suggests that Clean Zero might be ozonated water, i.e. pure water mixed with small quantities of ozone (O3). Ozonated water is an actual substance that can be produced on demand from air and purified water and used as a disinfectant.
However, ozone is an unstable and highly reactive substance that decays quite rapidly in water. Even under ideal circumstances (cold and very pure water) the half-life of ozone in water is only about 30 minutes, meaning that only 50% of the original ozone remains after 30 minutes, only 25% after 1 hour, only 6.25% after 2 hours and so on. Further, if the water is warmed up to room temperature or contains trace impurities that can react with the ozone, the half-life reduces to just a few minutes.
This is blatantly incompatible with Clean Zero's assertion that their product has an active shelf-life of several months. In particular, the Clean Zero website's page for their 500 ml plastic "Ready To Use Sprayer" bottles says (emphasis mine):
When required, simply fill directly from your Clean Zero 20, Clean Zero 80 or Clean Zero Hi-Vol unit and use on surfaces such as worktops, mirrors, washroom fittings and touch points. Store away from direct sunlight and your Clean Zero will remain active for a minimum of 10 months.
If Clean Zero is ozonated water, there won't be any detectable ozone left in it after storing it in a plastic bottle for 10 months (or even 10 hours), whether in direct sunlight or not.
It could be hydrogen peroxide.
Dilute solutions of hydrogen peroxide (H2O2) in water are also commonly used as disinfectants and they can be produced from oxygen and water using electricity and a solid catalyst.
While hydrogen peroxide is also unstable and naturally decomposes back into water and oxygen, it is significantly more stable than ozone. Under ideal conditions dilute hydrogen peroxide may decay by less than 0.5% per year but its decay can be accelerated by various factors, including both high temperature and the presence of catalytic impurities as well as exposure to air and to ultraviolet light (including sunlight). When stored in an unopened bottle and away from sunlight, commercially available hydrogen peroxide solutions can remain effective for up to 3 years, but opening the bottle and exposing the solution to air can reduce its effective lifetime to just a few months.
Of course, all this depends on both the original composition of the peroxide solution (including whether it contains any chemical stabilizers that slow down the decay), the way it's stored and — last but not least — on what you define the minimum effective concentration of remaining peroxide to be. Still, just as a random data point, this commercially available 1.4% hydrogen peroxide disinfectant spray is advertised as having a shelf life of 24 months, which is comparable to the 10 month active life advertised for Clean Zero. The difference between 10 and 24 months may plausibly come down to different initial peroxide concentrations, different storage conditions, different efficacy standards and/or the presence of stabilizer chemicals in the off-the-shelf product.
It might just be water (and snake oil).
All that said, I cannot rule out the possibility that Clean Zero is simply nothing but purified water. Indeed, their marketing materials seem to go to great lengths to suggest that their product contains nothing but pure "chemical free" water, somehow magically "embellished" to become an effective cleaning agent.
However, it's worth noting that just wiping a dirty surface with clean water (and a clean cloth) can be enough to remove a fairly large fraction of organic contaminants (including viruses, bacteria and other microorganisms) on it, especially if the surface itself is relatively non-porous and the dirt on it is not particularly oily or stuck to the surface. While water alone is not usually considered an effective disinfectant, I cannot rule out the possibility that, under sufficiently favorable testing conditions, it might show a measurable effect on ATP test results, which is what Clean Zero seems to be basing its efficiency claims on.
Update: The material safety data sheet for Clean Zero located by grafix, if confirmed to be accurate, pretty much confirms this option. The only substance listed in section 3 of the data sheet as being present in the product is distilled water. If ozone, hydrogen peroxide or any other reactive substance was included in the Clean Zero solution, it would legally have to be listed on the data sheet.
(FWIW, I very much doubt they would lie about the composition of their product on the official data sheet, since doing so would attract the attention of government agencies that take that kind of stuff very seriously and open them to legal action that would be very hard for them to weasel out of — much worse than, say, making dubious marketing claims.)