There are several requirements for a bicycle chain lubricant:
- It must penetrate to the innards of the chain very quickly -- this implies low viscosity
- It must stay inside the chain and lubricate the chain well for a long amount of time -- this implies high viscosity
- It must be flowing to be able to replenish (this requirement is not met by e.g. wax)
- It must lubricate the chain but this is not really an issue since even water can lubricate a chain -- until it evaporates away, that is, and water is excellent in causing rust
- It must not rust the chain
- It must not collect dirt excessively
Chainsaw chains are rapidly moving so I assume in a chainsaw the action that moves the lubricant inside the chain is the motion of the chain. In a bicycle, you typically lubricate the chain, wait the oil to get inside, and then wipe the excess surface oil away. A chainsaw lubricant might require you to ride the bike for a mile or two for the lubricant to get inside the chain.
Because chainsaw chains are rapidly moving, the lubricant must be thick so that it won't be flown away. Thus the second requirement is met: in the low-speed bicycle chains, the lubricant won't be thrown away.
The third requirement is met: as an oil it flows back once displaced.
Also the fourth requirement is easy: as an oil it lubricates.
The fifth requirement is easy: oil won't rust metal.
The real problem with chainsaw lubricants is that it may collect dirt easier than proper chain oils. This is of course a requirement that is very hard to be met as proper chain oils too collect dirt -- there is no lubricant that won't collect dirt.
So the chainsaw lubricant may be a bit slow to flow from the surface of the chain to the innards of the chain necessitating changing the lubrication procedure (ride a mile or two before wiping away the excess instead of waiting for a minute or two before wiping away the excess), and it may collect a bit too much dirt. Other than that, it meets the requirements.
A proper thixotropic motorcycle chain lubricant is better. It is applied from a spray can and agitated before spraying. The agitation and spraying makes the thixotropic lubricant thin, so it very easily penetrates the innards of the chain. Then when you let it settle, it gradually thickens. Start riding the bike and it becomes thin again due to the chain motion agitating it. Stop riding the bike and it becomes thick again.