They were de jure (officially, supposedly, on paper, in theory) ruled over by the king.
There are many terms to express this meaning. From Merriam Webster on "in name":
used with a following statement to say that something is so by name or title but that is not the way things really are
Nowadays, the Latin term de jure (as opposed to de facto meaning "in fact") is still commonly used to express such a situation. Terms such as "officially" and "on paper" are also used.
In this context, Tolkien is telling us about the history of Hobbits. While the Kingdom of Arnor was still functioning, they were still subjects (noun, meaning 3) of its king, citizens of the kingdom according to the laws of the kingdom - but in reality, they did their own thing and cared little for the kingdom or the world outside of their own domain.