What kind of paint this is that this happens to (or a family of paints, or even all paints)? Is this common among paints?
Although it's not the only possibility this would be common to the great majority of waterbased paints1.
No other common paint formulations (in the West at least) would be readily soluble in an alcohol.... the other coloured coatings that might be used on something like this include all oil-based paints (which includes all true enamels), or a solvent/resin formulation as with coloured lacquers of various types, which aren't really affected by alcohol.
As you used an 80% spirit the remaining 20% is primarily water, so there is a small chance the paint is water-soluble but I think this unlikely. A quick test with plain water will show whether it is or not.
Note that as this is surgical spirit, and not just "80% isopropyl alcohol" for example, there are likely to be other ingredients. In the UK the primary one may be methyl salicylate, but it's present at such a low level that I think it's unlikely to have any significant solvent effect even if it were a solvent for dried paint2.
Is it safe to use simple soap and water on this toy instead (or the wood bits of it)?
Safe, very likely yes. Effective, not so much. Soap doesn't directly kill microbes, it aids in the removal of contaminants from the skin because of its detergent action, with the mechanical rubbing/scrubbing having the main effect. So to be effective at truly decontaminating a surface like this cleaning with soapy water would need to be prolonged and involve brushing/scrubbing, which obviously isn't viable and would soon lead to the paint being damaged.
(I would be concerned about the paint being a water based paint and coming off
Waterbased paints are almost all insoluble in their original solvent/carrier, if they weren't they'd be at risk of just being dissolved by sweaty skin for example.
Waterbased paints (and varnishes incidentally) are nearly invariably based on an acrylic of some kind..... including so-called latex paint in the US, which doesn't have a latex base as the name might suggest. Almost all wall paints and many gloss trim paints and 'enamels' that feature water cleanup, are an acrylic of some kind with or without some additional modifiers to increase wear resistance, waterproofing and other characteristics.
or the wood reacting to the water in some way).
Despite being painted the wood can still react to water to a degree because the majority of waterbased paints are porous. Hence why waterbased exterior paints may be referred to as "breathable", since they allow water vapour to pass through so that the wood underneath can't get waterlogged as can happen with e.g. the alkyd paints that preceded them for this purpose.
Brief exposure to water shouldn't have a harmful effect on the wood however, but prolonged exposure can so is best avoided.
Off-topic but anyway
As a closing comment, common household bleach in solution with water is more effective, and lots cheaper, as an all-round surface disinfectant than alcohols. And note there's a marked difference in effectiveness of ethyl alcohol (ethanol) versus isopropyl alcohol or methanol.
NCBI page, Use of disinfectants: alcohol and bleach.
CDC source, Chemical Disinfectants.
1 What we call waterbased paints are emulsions (more accurately suspensions) in water of microscopic particles of something that isn't actually water soluble.
2 Even a very much stronger solvent than any alcohol is far less effective when heavily diluted with water. Acetone for example when diluted with water, as in some nail-polish removers, has a noticeably reduced solvent action compared to pure acetone.