No, the verse does not discuss an organ; it discusses a process.
In classical exegeses of the Qur'an, you will find that the verse does not discuss the place at which milk leaves the body of animals. Ibn 'Ashūr, in his tafsīr, explains that literary devices used in this verse as well as its intended meaning:
وموقع من بين فرث ودم موقع الصفة ل لبنا، قدمت عليه للاهتمام بها لأنها موضع العبرة، فكان لها مزيد اهتمام، وقد صارت بالتقديم حالا
NOTE. My own translation, so treat with care.
And the position of intermediacy of pomace and blood is descriptive of the milk, introduced earlier to draw attention to its intended meaning, which gives it more attention as when introduced earlier [in the sentence], it becomes a status [gramatically].
Ibn 'Ashūr further elaborated that milk was put as the object (maf'ūl, Arabic: مفعول) of "give you to drink" (Arabic: نسقيكم) to signify that the verse is not referring to a place of serving (i.e., udders), rather; to an outcome product based on two agents: cud and blood. The milk goes from "within" (butūnih, Arabic: بطونه) as excretion or blood would, but it is tasty and easy to swallow.
Note that the word used is butūnih, which is in a masculine form, whereas the word an'ām (animals specified in the verse) is in a feminine form, which would lend itself to using the word butūniha (feminine) rather butūnih (masculine). Linguistically, the origin of the word is batan (Arabic: بطن) has multiple meanings as is the case with most words of the Arabic language:
- What is within a place as in the case of Surat Al-Fath 48:24 referring to batn of Mecca (obviously, this is not referring to a physical belly of Mecca in the anatomical sense).
- What is within a body as in the case of Surat As-Saffat 37:144 when narrating the story of Prophet Yūnus within the whale's stomach or belly.
- What is on the outside lower side as in the case of Surat An-Nur 24:45 when describing animals that move on their bellies.
The word batan (Arabic: بطن) in its root format refers to what is below and concealed, with its antonym being dhahar (Arabic: ظهر) referring to what is above and apparent:
وَلَا تَقْرَبُوا الْفَوَاحِشَ مَا ظَهَرَ مِنْهَا وَمَا بَطَنَ
And do not approach immoralities — what is apparent of them and what is concealed.
This is often used as a literary device in the Arabic language and has often caused confusion when translating such words as batan can also mean belly (or bowel) and dhahar can also mean back.
Such usage was discussed by Sami Wadī' in his book At-Tafsīr al-Bayāni that is dedicated to the understanding of the linguistic and theological meaning of the words used in this chapter (not specifically verse 66, but it is covered too). He explained that the word farath (Arabic: فرث) referred to pomace. In the case of this verse, it is referring to the point of the process starting with the cud in fermented ingestion to the point where nutrients are transported through blood, and how milk — which neither resembles pomace nor blood — is formed. He, too, did not see that the verse referred to an organ, but to a process.