Unions are political rather than economic entities. Descriptions such as cartels or monopolies do not really apply.

Supporting other unions' strikes has always been an integral part of the labor movement. Sometimes this is done with sympathy strikes. They are normal strikes, and all the usual rules regarding strikes apply. The people on strike lose money, which usually keeps excessive sympathy strikes in check.

The American way is having a union contract that states it explicitly that employees do not have to cross picket lines. When there is a strike somewhere, other companies may stop deliveries and other services to the affected company, because they don't have any employees who could provide the services. Those employees are still getting paid, because they are working normally and doing everything they were hired to do. Their employers just cannot provide the services expected from them.