Is it really the case that children are showing brand preference, preferring one brand over others when presented options, because of a specific affinity for that brand?
According to research, it does seem that way.
A longitudinal study of 20 years (Guest, 1964) found that brand preferences originally expressed during the ages of 7–18 correlated with brand usage reported 20 years later, with the average amount of agreement between 1941 preferences and 1961 use being 23%. Guest concluded that early childhood experiences exert considerable influence upon later brand purchasing behaviour.
Your belief that such children aren't showing a brand preference, but are simply not familiar with the other products.
Ji (2002) whilst talking about the effects of brands such as Barbie and McDonalds on children, pointed out that parents, mass media and peers do play a part in it all:
Parents are probably the most instrumental in teaching children basic rational aspects of consumption, such as satisfying basic consumer needs in the marketplace (Riesman, Glazer, & Denny, 1956), understanding price–quality relationships (Moore & Stephens, 1975), handling money wisely (Marshall & Magruder, 1960), and shopping for different qualities among products (Ward, Wackman, & Wartella, 1977). Although parents are considered the primary socialization agents of children, no other agent of consumer socialization has received more attention (in the literature) than the mass media (Moschis, 1987; John, 1999). Through massmedia, including both advertising and editorial/program content (O’Guinn & Shrum, 1997), children may learn about new brands and products (Goldberg, Gorn, & Gibson, 1978), how to use products and who uses them (Atkin, 1978), realities and beliefs about them (Gorn & Florsheim, 1985), and preferences for them (Gorn & Goldberg, 1977). Peers also have input to children’s consumer socialization process. Studies show that young people learn the symbolic meaning of goods or expressive elements of consumption from their peers (Moschis & Moore, 1982). In addition, peers play an important role in the development of children’s preference for stores (McNeal, 1964), products (Hawkins & Coney, 1974), brands of selected products (Moschis, Moore, & Stanley, 1983), and things to collect (Baker & Gentry, 1996).
The thing is, a great deal of people do stick with what they know, and you could say this is in part due to people's comfort zones. Whether this phenomenon is the cause of people favouring certain brands is open to opinion. What is definite is that 100s of billions of dollars in business expenditure is spent looking at how they can use leverage such as television advertising, product design and even packaging, to encourage people to buy their product (Scheybani, 2015; Quelch & Cannon-Bonventre, 1983; Spanier, 2018). As well as with children, this affects adult behaviour, which as pointed out by Ji (2002), is passed onto children who in turn influence their peers through peer pressures.
Your title question: Do children really show brand preference at age 2?
Ji (2002) wrote (emphasis mine):
Why Study Children’s Relationships with Brands?
Studying the child–brand relationship phenomenon has its practical
implications. First, children consume a wide range of products from the
time they are born. For example, parents use J&J baby powder on their
babies, feed them with Gerber baby food, and cover them with Huggies
diapers. At around age two, children begin to eat their first solid food,
often cereal (Cole & Cole, 1996), and develop relationships with certain
brands, such as Cheerios, that may last a lifetime. As children grow
older, they not only consume more brands such as McDonalds’ food and
Barbie dolls through the purchases of their parents, but also begin to
make purchases using their own income (McNeal, 1999). In all, children
consume a wide range of brands and these early experiences may serve
as definite influences on their choices in adulthood (Guest, 1942).
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