Is the saying "You are forbidden to me like my mother's breast" considered a divorce, especially if uttered in a moment of anger without the intention of prohibition? What is the ruling of the religion on this, and is expiation obligatory, and how is it distributed?

If by "breast" you meant milk, and you likened yourself to it, then this is not considered anything [problematic], because mother's milk is not forbidden to her son after weaning. However, if you meant the breast itself, or if you had no specific intention, then the prohibition in this case is dhuhur.

Scholarly Opinions ( Madhahib ): Shafi'is and Hanbalis: They consider likening one's wife to a limb of the mother other than the back – such as the hand, head, or breast – to be explicit dhuhur, especially if the limb is not mentioned for honor and respect. Hanafis: They do not consider this dhuhur, as they stipulate that the likened limb must be forbidden to look at, and they permit a woman's mahram to look at her breast. Malikis: For them, explicit dhuhur is limited to likening one's wife to the back only; anything else is a metaphor (kinayah) and does not constitute dhuhur unless intended. However, if the word "unlawful" (haram) accompanies the phrase, then it becomes dhuhur.

Conclusion: According to the majority of scholars, you have committed dhuhur, and you must perform the expiation for dhuhur, which is in order of precedence: freeing a slave, and if you cannot find one, then fasting for two consecutive months, and if you are unable to do that, then feeding sixty poor people. You must also repent to Allah for this act.