Both the Bhagavad Gita and the Analects of Confucius are about duty. They argue that individuals have social responsibilities that take precedent over personal preference and even personal principles.

The Bhagavad Gita is a dialogue between humanity, represented by Prince Arjuna, and the eternal, represented by the divine Lord Krishna. The subject at issue is duty, or more accurately, dharma. "Duty" is not a perfect translation of the Sanskrit word dharma, but it's close: dharma is the behavior required to maintain "rta," the good order of the universe.

The conflict at the heart of the Bhagavad Gita is between personal principle and dharma. Arjuna is about to go to war with the Kauravas. He has relatives, teachers, and beloved friends among them. His personal principle tells him, as it would tell anyone, not to fight. Dharma, in the form of Krishna, tells him otherwise. It reminds him that human perspective is limited and that, though a great hero, his own life is ephemeral. There are greater things than Arjuna, says Lord Krishna, and dharma, the right way of living that serves that greater order, is more important than Arjuna's own feelings.

The Analects makes the same argument directly, without the dialogue. One of the most important concepts in Confucian thought is "li." Like "dharma," it doesn't quite translate to English, but a good start might be "good order." Li represents the concrete behavior expected from people of every kind, fulfilling every role in society that makes that society run smoothly (see reference). The Analects present thoughts on how li may be achieved and call on the reader to behave accordingly, just as the Gita presents a dialogue about dharma and calls upon Arjuna, and thereby the reader, to fulfill it.

Despite being separated by hundreds of miles and radically different cultures, there is a thread of unity between the Bhagavad Gita and the Analects. Both acknowledge the existence of an order beyond worldly life, and both ascribe the highest moral value to following it rather than personal inclination.

Both the Analects and the Bhagavadgita focus on ethical behavior.

Confucius did not focus on a higher being, but he did propose that each person should be his best self.  The Bhagavadgita suggests that ethical behavior comes from proper spiritual development.

Confucius proposed that society benefited most from ethically behaving individuals.

He believed that individuals could begin to cultivate an all-encompassing sense of virtue through ren, (Wikipedia)

The Bhagavadgita also proposes that a person must be spiritual to be ethical.  A strong spiritual being is the key ethical behavior.

O Arjuna, one who in this world does not apply the procedures prescribed and established by the Vedic scriptures; that person living in sin wastes their human life captivated by sense gratification. (http://www.bhagavad-gita.org)

Each text has ideas about living ethically, and each proposes spiritual development as a method of maintaining ethical behavior.  The Bhagavadgita seems to rely more on spiritual procedures to get there.

How are the Analects and the Bhagavad Gita related in their intentions and thoughts?

Both of these works involve a master and a disciple. In both works, the master gives instruction to the disciple and answers the disciples questions. In the Bhagavad Gita, Krishna is the master and Arjuna is the disciple. Both works are concerned with how an individual fits into the larger society through their relationships. In both works, the master instructs the disciple to honor their duties and social responsibilities. The Bhagavad Gita and the Analects give special attention to the parent–child relationship, the teacher–student relationship, and the king–subject relationship. The works emphasize that a well-functioning society depends upon socially harmonious relationships.

Both works also discuss the art of ruling. The master gives instruction to the king and governing lords. Both works explore themes of justice, fairness, virtue, and righteousness. The aim of both the Analects and the Bhagavad Gita is to help the disciple grow into a just and honorable ruler.