A “milkaholic” (milk + -aholic, related to “alcoholic”) is a person who is addicted to drinking milk. “Our daughter, age two, is a ‘milkaholic’” was cited in print in 1955. The term originally referred to an infant who took milk and refused all other food. The term “milkaholic” soon enlarged to encompass an individual of any age who consumes a lot of milk.
 
   
Wiktionary: milkaholic
Etymology
milk +‎ -aholic.
(...)
Noun
milkaholic (plural milkaholics)
1. One who likes to drink milk, particularly an excessive amount.
 
(Oxford English Dictionary)
-aholic, suffix
orig. U.S.
Forming nouns (often humorous nonce-words) denoting a person who appears to be addicted to the thing, activity, etc., expressed by the first element, as computerholic, newsaholic, spendaholic, etc.
1955   Hutchinson (Kansas) News-Herald 28 Dec. 14   We must face it. Our daughter, age two, is a ‘milkaholic’.
 
28 December 1955, Hutchinson (KS) News-Herald, pg. 14, col. 2:
What Do You Do With A Little
Girl Becoming A Milkaholic?
By BILL BORK
We must face it. Our daughter, age two, is a ‘milkaholic’.
 
A milkaholic, you see, is a little on the order of an alcoholic. The basis of the ailment—either type—is supposed to be a vague emotional disturbance that causes the victim to seek relief in the bottle.
 
2 October 1962, The Arizona Republic (Phoenix, AZ), pg. 26, col. 2:
Milkaholic Worst Kind
DARLINGTON, England (AP)—A grocer has been fined for selling a mother milk for a baby after hours, but if he had sold her an alcoholic drink he would not have violated the law.
 
23 September 1963, Aiken (SC) Standard and Review, “Potpourri” By Helen Herre, Teen Times sec., pg. 1, col. 3:
The new club at A.H.S.—milk-aholics anonymous!
 
Google News Archive
5 May 1965, Milwaukee (WI) Sentinel, “Child Care” by Drs. William G. Crook and Walton W. Harrison, pt. 3, pg. 5, col. 1:
Your baby is a “milkaholic.” He has been drinking an excessive amount of milk over a long period of time.
 
8 June 1967, Marietta (GA) Daily Journal, “Child Care: Get ‘Milkaholic’ Away from Bottle” by William G. Crook, M.D., pg. 14, col. 7:
I’m not sure that we doctors understand just what happens to babies like your youngster, but doctors around the country describe them as being “milkaholics” or “milkomanics.”
 
Google Books
Dairy and Ice Cream Field
Volume 156, Issues 7-12
1973
Pg. 14:
“When I’m thirsty, and that’s most of the time, I drink milk,” Bill says, “I guess you might class me as a “milkaholic.” the acid and bitter taste was present but to a lesser degree than when made by the normal process.
 
Google News Archive
7 June 1997, The Deseret News (Salt Lake City, UT), “Ailing milk-a-holic suing dairy industry” (AP), pg. A2, col. 2:
A self-described milk-a-holic is suing the dairy industry, claiming that a lifetime of drinking whole milk contributed to his clogged arteries and a minor stroke.
   
Google Books
Save Our Sleep: Toddler
By Tizzie Hall
Sydney: Macmillan
2010
Pg. 42:
Milkaholics
Milkaholics are toddlers over thirteen months whose diet is mainly milk-based. This often results from a late introduction of solids.
     
Twitter 
Jonnie Arnold
‏@ayyyylmao_
Welcome to MA-Milkaholic Anonymous
4:02 AM - 14 Aug 2014
 
Scoop (New Zealand)
Kiwi Kids Need Red Meat
Thursday, 14 August 2014, 4:24 pm
Press Release: Beef And Lamb NZ
(...)
Fiona Greig, Nutrition Manager of Beef + Lamb New Zealand comments on the implications of iron deficiency in young ones. “Dietary iron is an essential mineral required for normal growth, development and cognitive function in young children. As toddlers are picky eaters, nutrient dense foods are crucial at this critical time.“
 
“As this study highlights, many toddlers are filling up on too much milk or formula, which are often coined as ‘milkaholics’, filling up their little tummies and leaving little room for a diverse range of foods including those rich in iron” says Greig.