I am answering this wrt cows milk, not milk of other animals like goat, camel etc.
As stated here in SB 8.8.11:
Pañca-gavya, the five products received from the cow, namely milk,
yogurt, ghee, cow dung and cow urine, are required in all ritualistic
ceremonies performed according to the Vedic directions.
So it's an important ingredient to perform vedic rituals. Also cow milk was meant for consumption as cows produce more milk than it needs to feed it's calf but humans are to consume the milk only after the calves had their share.
Quoting from an article posted on bhaktivedantamanor.co.uk here:
A point of consideration is that milk is a food sanctioned and
designed by God, by Krishna, for humans and not just for the calf. The
cow produces more milk than the calf needs and this is not accidental
but by design. To get milk from a cow you need to impregnate the cow
(there are numerous examples where some cows gave milk without
impregnation but that is another story) and after a pregnancy of nine
months a calf will be born and the cow will produce milk. Out of
affection for her calf the cow will give as much milk as possible. The
first milk is full of colostrum and this will give the best start to
the calf. After about five days the milk looks normal and this is then
suitable for us to drink. The milk is for the calf and for us.
Read the purpose of Surabhi (cow) as explained here SB 8.8.2:
These cows are the Lord’s pet animals. From the surabhi cows one can
take as much milk as one needs, and one may milk these cows as many
times as he desires. In other words, the surabhi cow can yield milk
unlimitedly. Milk is necessary for the performance of yajña. Sages
know how to use milk to elevate human society to the perfection of
life.
Citing another quote to support the above statement Satapatha Brahmana Part V (SBE44):
14:2:1:99. He then lets the calf to it (to suck), with, 'Pûshan thou
art,'--Pûshan, doubtless, is he that blows here (the wind), for that
one supports (push) all this (universe); and the Pravargya also is
that (wind): it is him he thus pleases, and therefore he says, 'Pûshan
thou art.'
14:2:1:1010. He then leads it (the calf) away with, 'Afford (milk)
for the Gharma!' for the Gharma, doubtless, is that fluid which this
(cow) lets flow: he thus means to say thereby, 'Allow her a share!'
