Is it considered a breach of an oath for a man to give his wife's phone number to other parties after he swore, "By divorce, I will not give it to anyone," due to compelling circumstances, knowing that his wife later contacted her family without his knowledge, and what are the consequences of that?

The preponderant view regarding swearing by the forbidden is that if divorce is intended by it, then it is a divorce. If ẓihār (a form of divorce by injurious assimilation) is intended, it is ẓihār. If an oath is intended, or nothing specific is intended, then it is an oath.

If you did not intend divorce or ẓihār, then you owe an expiation for an oath (kaffārat yamīn) upon breaking it.

Your breaking of this oath depends on your intention: If you intended never to give your wife's phone number to anyone at all, then you have broken it. However, if you only intended not to give your wife's phone number to her family, then you have not broken it, because intention specifies the general.

Swearing by the forbidden is not permissible. The permissible oath is swearing by Allah Almighty.