I found a solution after lots of debugging to understand the internals:

VertxContext.getOrCreateDuplicatedContext(_vertx).executeBlocking(  
    () -> {
        MDC.put(xxx);
        operation.accept(tenantId);
        logger.info(xxx);
        return null;
    });
    

VertxMDC creates a duplicate context from the current context unless the current context is already a duplicated context.

Without getOrCreateDuplicatedContext(), the currentContext inside the operation is the EventLoop context (in my case), so every MDC call (put, get) creates its own duplicated context and stores/retrieves data in a storage that no one else will ever use.