This presentation shared by A Rogue Ant suggests some insights. In particular:
- Children exposed at a young age (starting at about six months seems to be the critical period) to a wide variety of food are more likely to be receptive to a wider variety of foods when they are older.
- Culinary "growth" seems to start around seven years of age.
From this, we can surmise that, while there are biological aspects at work, they can be shaped to a significant degree by early experiences. Put differently, it would seem that a child's experiences from about six to nine months may largely dictate the child's preferences for years. Thus, a child given "adult" food very early and consistently stands a very good chance of accepting such food throughout childhood and beyond, while a child given only "mild" food in the first year is likely to develop "fussiness" which may last until adolescence or even persist for life.
Strictly speaking, then, "culture" plays a significant role, but to be more precise, it is a child's experiences (i.e. exposure to various foods) in this "critical formative period" that is most relevant. This has, of course, a significant cultural aspect, but it would be well to keep in mind that after this period, a cultural / environmental change, at least from "simple" foods to "complex" foods, would seem unlikely to be "effective".
What about non-humans? Well, it seems plausible they would be in the same boat, at least inasmuch as appropriate cultural conditions can "circumvent" the stereotypical "picky child eater". I'm not aware of any instances of young animals having significantly different "tastes" that develop and expand as they grow older (although documentation to the contrary would be welcomed!), and on some reflection, this seems to make sense. In the wild, young animals don't have the luxury of being fussy eaters, as the only food available is whatever mom and dad are eating. Thus, they would naturally be exposed to "adult" food at an early age, and we might presume the same principles at work in humans are at work for other animals as well.
However... an interesting alternative take is also available. Domestic cats have a reputation for being fussy eaters. Being raised by humans, they are more likely to be exposed to only a limited diet during the relevant formative period. Combined with the idea that domesticated animals are in some ways developmentally stunted, never "fully" maturing to adults, this may well be a case of the same principles manifesting, with a limited diet early on leading to less tolerance later. Thus, it seems quite plausible that non-humans might demonstrate a similar spectrum of fussiness or lack thereof based on experience during the critical formative period.
So... TL;DR, both humans and non-humans might believably tolerate or not tolerate "adult" food as children.