The key to knowing what is in a tube of paint isn't the generic or common name given to the color but rather the Color Index Name or pigment code and number. Exactly where this is on a paint tube label varies from brand to brand, but any decent paint will have it.
A Color Index Name starts with one of 10 pigment codes from the Color Index, for instance PB (Pigment Blue), PR (Pigment Red), or PY (Pigment Yellow). This is followed by a number for a specific pigment. Every different pigment used for paint has a different Color Index Name.
For instance PB 29 is Pigment Blue 29. This pigment is generally sold as French ultramarine, so if you're looking at a tube labeled French ultramarine you can check if it contains PB 29 or not. If it does, it should be near identical to the color you're familiar with.
If a paint color seems to be available in two versions, one with the word hue after it, it's likely they're made from different pigments. The hue version is usually made from cheaper pigments, though sometimes it's modern-day equivalents of old pigments that may not be lightfast or are poisonous.
If a paint is a cheaper or student's quality brand, extenders or cheaper pigments may be added to stretch the more expensive pigments. The tube label ought to tell you if another pigment has been added, whether the color is a mixture of pigments (though some cheaper brands don't).