"Know" has two meanings: be consciously aware, and have some account of that fact in your cognitive apparatus. And then, "things like that" really needs another thing like that plus a third thing that is not like that. I'll supply some examples for you – "boil" and "bullion" (also like), and "dog" / "dogs" (not the same kind of thing).
The word "Jupiter" might be known to a reasonable number of under-10 children, clearly not a majority, and all would be educated. Probably only a handful know "Jovian". So virtually none of the set you're interested in know that relationship. Likewise with "boil" and "bullion", where "boil" is probably known to most English speaking children between ages 4 and 10 (but not "bullion"). Virtually all children know "dog" and "dogs", and connect them instantly.
Very few children would understand the technical terms, so you can't ask if Jovian is the adjectival form of Jupiter or probably even if "dogs" is the plural of "dog". Most people don't talk about "adjectival forms", "plurals", so uneducated English speaking children at age 10 probably have no idea what a plural is. Even educated ones struggle, since grammar isn't taught so much.
Whether or not this is part of one's subconscious linguistic ability is hard to tell in the cases that you're interested in. There is a classical test, the wug test, that shows that children have in fact internalized regular plural formation. As far as I know, nobody has even attempted to do the same test with words like "Jovian" and "sangiovese" or "Jupiter".