I'm wondering if you use a spray bottle, not an aerosol can, to spray a cleaner containing mostly water and ionized surfactants into the air, will this remove odors from the air? I know things like Febreze and baking soda have molecules that bind to odor-causing molecules so that we don't smell them anymore, but I haven't been able to find a discussion yet on whether ionized surfactants work in a similar way to these molecules.
First of all the mechanism does not affect the chemical interactions. So, if an atomized spray is effective it doesn’t matter whether it was produced by pressurized gas in an aerosol can or a manual pump in a spray bottle.
Secondly, there is a difference between masking an odor with other odors and removing it by adsorption or reaction. Most odors are removed by adsorption because they share physical properties with similar adsorption characteristics. They usually aren’t “reacted” with anything because odiferous molecules just come in too many different kinds for a particular reaction to be universally useful.
Finally, whether a particular ionized surfactant (and I am not clear whether you mean a surfactant that has been ionized, or what we call an ionic liquid) binds to an odor molecule in any useful way or not, he two previous conditions apply. Any suitable applicator will do the job. Any particular “ionic surfactant” may or may not be suitable - a general statement about them and odor removal cannot be made.
I also am somewhat confused as to what you mean by "ionized surfactant". Surfactant molecules come in basically 4 different types: anionic, cationic, nonionic, and zwitterionics. These terms describe the form the molecule assumes when it is added to water: anionics become negatively charged (with a positive counter ion like sodium), cationics become positively charged (with usually a chloride counter ion), nonionics remain uncharged, and zwitterionics have both negative and positive charges. All surfactants (except nonionics) form ionized or charged forms when added to water.
Surfactants in water form assemblies called "micelles" if the concentration is high enough. The "critical micelle concentration" (the concentration at which micelles are formed) is usually much less than 0.01% so essentially any surfactant/water mixture that is over 0.01% has micelles. In cleaning situations it is the micelles that capture grease and oils within them and carry them away in the cleaning solution. Usually the material trapped in the micelle is of an oily or hydrophobic (water repelling) character.
The products like Febreze that you mention are formulated with a molecule named beta-cyclodextrin. This is a molecule with a cage like structure. The odor molecule can be absorbed into the "cage" of the beta-cyclodextrin and becomes trapped. According to the patents that have been published some polymers are also used. Polymers are very large molecules that are capable of wrapping around smaller molecules. These also interact with odor molecules to decrease their volatility (ability to evaporate). The patents also indicate that small amounts of surfactants are used in Febreze type products, but they are there to make sure that all the components of the product stay mixed with the water. There are a fair number of criteria they use to choose the surfactants but according to the patent it is the cyclodextrin or the polymers that do the odor trapping, not the surfactants.
If you're interested you can read one of the Febreze patents to see how it is put together: https://patents.google.com/patent/US9260817B2/en
For what you are proposing to work, the odor molecules would have to be able to be trapped in the surfactant micelles in a droplet of spray. A single surfactant molecule would be unlikely to "bind" with the malodor molecule - it would be more likely that a micelle assembly would do trapping. The idea that the surfactant is "ionized" does not determine whether or not it could do this because even nonionic surfactants (which do not ionize in water) form micelles. As Steven said it is not able to be predicted whether any particular surfactant or mixture of surfactants could accomplish this, since there are many different types of micelle assemblies with different affinities for different molecules. As I said before the patents I was able to look up all contain surfactants but they are still adding cyclodextrin and polymers to be the odor trappers. Polymers are more expensive ingredients than surfactant so if they could do this with surfactants they would probably not be using the other ingredients. It is possible that a spray solution of surfactants might decrease malodor but whether it could eliminate malodor as efficiently as Febreze claims is doubtful.
From what I found on the internet, baking soda (which is a basic or alkaline compound) interacts with odor molecules that are acidic reacting with them in an acid-base neutralization which will change the smell to something more mild or completely without odor. This is an example of what Steven told you - there are so many different types of odor molecules that if an odor is eliminated by a reaction it only works with a subset of molecules and is not broadly useful. This also means that baking soda has a completely different way of eliminating odors than Febreze in that does not involve trapping or binding odor molecules.
I would also caution you against making a fine spray of a surfactant solution. If you sprayed a surfactant solution and it made a fine mist those droplets would remain airborne for a long time; if you then inhaled those droplets they might be irritating to your nose as surfactant solutions can be irritating to mucous membranes.
I hope this supplementary information is helpful.