Bottled Betadine is a skin disinfectant and it is typically on drug store shelves among first aid supplies. Nasal irrigation bottles, distilled water and packets of powder to make up saline are also on the shelf (e.g. NeilMed Sinus Rinse kits), typically among cold/flu treatments.
Saline nasal irrigation can be soothing and help clear gunk. Some doctors recommend adding Betadine to the solution for anti-viral activity. You'd need to do a little research on appropriate concentrations if you decide to try this. Excess iodine can be irritating and shouldn't be swallowed. It's also important that any water used to make up nasal irrigation fluid be sterile. You don't want to squirt tap water up your sinuses unless you especially enjoy sinus infections.
I think it is usually 1% concentration which is recommended. This article recommends 0.6%
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8026810/
This is a link to an abstract with other articles cited.
https://pubmed.ncbi.nlm.nih.gov/32520599/
Hydrogen Peroxide as a nasal spray,
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7262503/
I have heard that hydrogen peroxide can be substituted for the iodine for those allergic to iodine.
There are several different bottled Betadine products. One is a skin disinfectant (usually in red colored packaging) as you point out. Another is a premade nasal spray and they also have a gargle product (both in blue packaging). Folks should look carefully to see which they are purchasing.