Is there a difference in what oil you put in the crankcase vs the fuel mixture?

Sounds like broken or stuck piston rings, allowing the air / fuel mixture during the compression stroke, to go past the rings and into the crankcase, which pressurizes and forces the oil out thru the dip stick. good luck

If I understand your question correctly, the liquid is coming out from the oil cap / dipstick. If this is the case, DO NOT TRY TO START THE MACHINE, as there is a risk of EXPLOSION!!!
This fault is most likely caused by a flooding carburettor. If the float needle is not sealing correctly and the fuel tank is above the level of the carburettor, fuel will continue to flow through the carburettor and into the engine cylinder. It then makes its way down past the piston rings into the crankcase / sump. It may also travel into the engine via the crankcase breather tube.
To confirm this fault, smell the oil.
If it smells like fuel, then this fault is confirmed and you must have the carburettor serviced. You would then need to drain the engine oil and clean the spark plug. Before you re-fill the engine oil, leave the filler cap off for 6 hours, but cover the hole with a rag. This will allow fuel vapours to dissipate.

First check to see if the oil level in your crankcase is too high. If you have a vacuum petcock malfunction (hole in the diaphragm) or if the fuel petcock is left in the prime position and a float needle sticks, then you get you get gas running into the cylinders, past the rings, and into the oil.
When you start your motorcycle the oil/gas mixture has no place to go, the crankcase is too full, and the mixture exits through the crankcase breather into the air box where it will ruin a paper air filter. Always turn your fuel petcocks off when not driving. If the petcock diaphragms are bad rebuild the petcock with K&L rebuild kits or replace the fuel petcocks with new ones. If the float needles in the carburetors are bad or sticking, take the carburetors apart, clean them, and replace any worn or damaged parts You will have to change your gasoline saturated oil for it will no longer properly lubricate your motor.

Your Honda would be a four stroke engine, so your fuel would be straight gas, you will notice on the base of the engine there is a location to put oil, a 4 stroke engine is lubricated by the oil in the base of the crankcase. On a two-stroke engine.oil fuel mixture lubricates the crank so it has no provision to add oil to the engine.

When the float in the carb sticks, it allows fuel to drain into the cylinder, that runs pass the rings into the crankcase. When the crankcase is full the pistons cannot go down, and the starter won't turn the engine. Drain all the oil gas mixture, and refill with proper oil. Replace float, needle and seat in the carb bowl. Throughly clean fuel passages, and bowl. If motor won't turn, remove spark plug and turn over engine a few times to remove gas oil mixture from the cylinder too.

Sounds like your rings are either stuck or broke. Check the compression, it's probably low or non existant on at least one cylinder. In either case it sounds like a tear down.

A 4 stroke motor does NOT have Oil in the Fuel. There is NO fuel mixture. Just use regular gas. The only place you need oil is in the sump, and regular 20/30 motor oil will be fine.This only needs to be replaced when low on dipstick in crankcase.

Blue exhaust smoke means one thing: crankcase oil getting into the combustion process. The most likely cause is worn piston rings. Other possible causes are worn valve guides or a blockage of the crankcase vent. Tip: blue exhaust smoke under acceleration = piston rings. Under deceleration = valve guides.
Also make sure someone did not put a gas/oil mixture in the fuel tank by mistake (thinking it had a two cycle engine).

I'm gonna go with theory #2 sort of. My guess is that one or more of the float valves is leaking causing the gas to overflow into the crankcase. This happens often with bikes that are allowed to sit for long periods with gas in the carbs and tank. A thorough fuel system cleaning is what it needs. I'd say go ahead and buy four carb rebuild kits for it - may as well since you'll have to disassemble the carbs to clean the gum and varnish out, which is really the best thing for it at this point. I wouldn't try to run it again 'til you get the carbs to quit leaking, the fuel drained out of the crankcase and put fresh oil in. ,,,

Change your spark plug
If the fuel oil mixture is too oily it will foul the spark plug and no cleaning will get it right.
If there is a lot of oil in the crankcase it may foul the new spark plug so another new one is needed