A league is a group of people (fx. parties (political), states or athletic teams) who've come together because of a shared interest.
An alliance is an agreement to work together for a specific purpose.
A union is also a group of people/states/companies who've come together because of a shared interest or purpose but simply putting two or more things together would be a union, like fx. a marriage would be a union or merging north and south korea would be a union.