Well, for starters, homeopathy means that they dilute the stuff so much that there’s actually barely a molecule left of the initial substance when they’re done preparing it. So there’s your answer for why this partucular product probably isn’t dangerous.
Lots of vaccines are made by killing viruses or bacteria so that they can’t actually infect the patient. They’re treated with solvents or heat until they’re just useless shells of their former selves, and the body produces an immune response aganst them without having them fight back at all.
Some use actual live organisms which have been carefully cultured to make them lose their dangerous (virulent) properties. Strains related to the disease-causing organism, but with little or no virulence themselves, can also be used.
When an organism’s toxic effects are from a toxin it produces, vaccines can be made by modifying the toxin slightly to make a toxoid, which looks enough like the real thing to cause a good immune response, but won’t do any damage.
So, in a sense, you can start with pus, or sputum, or any other pleasant infected bodily excretion you can get a swab into, plate it on the right media, treat it with the right chemicals, and get a working vaccine. Not quite as simple as mixing some pus into a solution and drinking it down, though.