This would create uncertainty in mahram relationships as a child would feed on the milk of multiple anonymous women, and a woman's milk will be consumed by multiple anonymous children, making it impossible to observe Islam's commands regarding what is Halal and Haram.
This is different from when parents hire a wet nurse, because they know which woman fostered their child and the wet nurse knows which children she fostered. So it is possible to identify mahrams when arranging marriages.
There is a relevant resolution on this by the Islamic Fiqh Academy under the OIC. The arabic text can be found here:
بسم الله الرحمن الرحيم
الحمد لله رب العالمين والصلاة والسلام على سيدنا محمد خاتم النبيين وعلى
آله وصحبه أجمعين
قرار رقم: 6 (6/2)[1] بشأن
بنوك الحليب
إن مجلس مجمع الفقه الإسلامي الدولي المنبثق عن منظمة المؤتمر الإسلامي
في دورة انعقاد مؤتمر الثاني بجدة من 10 – 16 ربيع الآخر 1406هـ الموافق
22 – 28 كانون الأول ( ديسمبر ) 1985م،
بعد أن عرض على المجمع دراسة فقهية، ودراسة طبية حول بنوك الحليب
وبعد التأمل فيما جاء في الدراستين ومناقشة كل منهما مناقشة مستفيضة شملت
مختلف جوانب الموضوع وتبين منها
أولاً: أن بنوك الحليب تجربة قامت بها الأمم الغربية، ثم ظهرت مع
التجربة بعض السلبيات الفنية والعلمية فيها فانكمشت وقل الاهتمام بها۔
ثانياً: أن الإسلام يعتبر الرضاع لحمة كلحمة النسب يحرم به ما يحرم
من النسب بإجماع المسلمين۔ ومن مقاصد الشريعة الكلية المحافظة على النسب،
وبنوك الحليب مؤدية إلى الاختلاط أو الريبة۔
ثالثاً: أن العلاقات الاجتماعية في العالم الإسلامي توفر للمولود
الخداج أو ناقص الوزن أو المحتاج إلى اللبن البشري في الحالات الخاصة ما
يحتاج إليه من الاسترضاع الطبيعي، الأمر الذي يغني عن بنوك الحليب۔
:قرر ما يلي
أولاً: منع إنشاء بنوك حليب الأمهات في العالم الإسلامي۔
ثانياً: حرمة الرضاع منها۔
والله أعلم ؛؛
There is an english translation here [p. 24] as well as on Islamweb:
Bismillah Arrahman Arrahim
Praise be to Allah, the Lord of the Universe, and Prayers and
Blessings be upon Sayyidina Muhammad, the last of the Prophets, and
upon his Family and his Companions
RESOLUTION N° 6 (6/2)
CONCERNING MILK BANKS
The Council of the Islamic Fiqh Academy, during its second session,
held in Jeddah (Kingdom of Saudi Arabia), from 10 to 16 Rabiul Thani
1406 H (22-28 December 1985) ;
After a legal and medical study have been submitted to the Academy,
and Having considered the contents of the two studies and having
discussed each one of them in depth covering the various aspects of
the subject ;
First: The Milk Banks concept and experience are a western nations
initiated phenomenon ; however, as the concept was being practiced,
some scientific and technical adverse effects were noticed, thus
causing decrease in its use and lack of interest in it.
Second: To Islam, Breast feeding creates a bond similar to a lineage
bond and forbids, according to the consensus of Muslim jurists,
exactly the same which is forbidden due to actual lineage
relationship. One of the goal of Shari'a is to safeguard the lineage
of a person whereas the Milk banks lead to mix up and doubt.
Third: The social structure in the Muslim world is such that it can
fulfill the needs of a premature or a weak child in need of natural
breast feeding from human milk (in some special cases), thus
eliminating dependency on the Milk banks.
Accordingly:
First: The establishment of Milk Banks should be prohibited in the
Islamic World.
Second: It is prohibited to feed a Muslim child with milk from these
banks.
Verily, Allah is All-Knowing