Every----is used to express "all". Every individual child. I don't suggest using it here because it would be untrue. Not "every" child eats sweets, even though we might expect that they would. For example, if we change the context of the sentence from eating sweets (a sensory experience) to learning (personal growth) it might be easier to use "like every": He learns through experience, like every child. Any---is less specific, used to express things that are more general. "like any child eats sweets"= children "in general" eat sweets, it's not uncommon. Hope this helps, feel free to contact me if you have any questions. :)

"He eats a lot of sweets like every child." 'Every child' means the total number of all children. "He eats a lot of sweets like any child." 'Any child' can mean 1 child or 5 children or all children. 'Any' is an uncertain amount of something. 'Every' is the total amount of something. For example: "Is there any rooms available this evening?" 'Any' meaning a small number of rooms. "Yes every room is available this evening" 'Every' meaning the total number of rooms in the hotel are available/empty. I hope this helps! :)

Any and every do have different meanings, however they can be used in such a way that they refer to the same thing In this case, they both mean that all children eat sweets. "Every child eats sweets' states directly that all children eat sweets. 'Any child eats sweets' implies the same thing: literally it means that whenever a child is chosen randomly he/she will be a sweet-eater: this is only true if all children eat sweets.

I just wanted to point out that although your sentence is grammatically correct, "He eats a lot of sweets like any other child." sounds a tab bit better / slightly more natural to me. Sorry to add another complication to this question. :)