the Union was an alliance of sovereign states

No, it is not, because the constituent states of the United States are not sovereign.

A sub-state is a territorial and constitutional community forming part of a federal union. Such states differ from sovereign states, in that they have transferred a portion of their sovereign powers to a federal government.

It was a league of friendship, in which the states were sovereign and the national government had only weak delegated powers.

Mexico is a state in the sense that it is an independent, sovereign nation. Its formal name is "the United States of Mexico."

Puerto Rico is not a state in the sense of sovereign nation because it is not sovereign. It is part of the United States of America. It is also not a state in the sense of being one of the states of the Union that is the United States of America, because it has not been admitted to the Union as a State. It is an organized territory of the US and has a large degree of local autonomy, but it is still a territory and subject to control of Congress and the President to a much greater degree than is one of the states of the Union.