The better term is miscible in all ratios. The idea behind the term is that you can have a mixture from 99.9999… % water and 0.0000…1 % ethanol to a mixture of 0.0000…1 % water and 99.9999…% ethanol. No infinity necessary.
The obvious opposite to miscible is immiscible (in some or all ratios). A common pair of household liquids that are immiscible are fatty ...
"How to verify that ethanol is infinitely soluble in water?".
You cannot do it because it is WRONG. And not because it is false for ethanol, it is simply plain wrong conceptually and mathematically.
Solubility is the ratio between the amount of the specified substance and the volume of the system.
As for a substance occupies its own volume, ...
Both water and ethanol are colourless. How can I observe that they are infinitely soluble ?
They have different refractive index. You will see Schlieren as you mix them, which will eventually disappear. If two solutions don’t mix, there will be a visible interface. If you have trouble seeing it, mix it and let it settle.