This is of course a big question and I don't believe that there is a definite answer to it. A very thorough investigation of this matter comes from Rogers and McClelland (2004), who have a developed a parallel distributed theory of acquisition, representation and use of human semantic knowledge. As the name implies, this effort comes from the realm of parallel distributed processing, which is also called connectionism. The books that @DikranMarsupial cited can be seen as the foundations of the field. As far as I understand, the two volumes and their authors (McClelland is only one of them, at least two other important names are David Rumelhart and Geoffrey Hinton) had an enormous impact on the revival of neural networks within cognitive science and on the development of the parallel distributed approach.
Rogers & McClelland (2004) offer an interesting account of how things are grouped into categories, which is what you are asling about. To give a very brief explanation: according to their theory, all objects have certain attributes. For example, a canary has feet, wings and feathers. A robin also has all these things. A pine, on the other hand, has leaves, roots and roots. The same is true for an oak. In reality, there are much more attributes, but this should be enough to make the point.
Objects that tend to covary with one another in a coherent way (because they have similar attributes) are perceived as similar and are classified as belonging to a category, like the canary and the robin. Other objects, that do not covary with those from this first group, but that do covary coherently with other objects, are classified as belonging to another group, like the pine and the oak. It is these coherent categories that are important for classifying objects.
Rogers and McClelland (2004) used a neural network to demonstrate and test their theory. Their findings are consistent with a range of findings from developmental studies. They also have an article, which is basically a short summary of the book (Rogers & McClelland, 2008). If you are interested, you might want to check out this first. The article has peer reviewers commentaries that point out weaknesses of the theory, which is also very interesting.
Since you refered to machine learning in your question: the learning algorithm used in the simulation studies was the backpropagation of error - algorithm. As was also mentioned by @DikranMarsupial, this is not an especially biologically plausible algorithm. Interestingly though, as is also explained in much detail in the book, it is possible to predict the order in which the objects within the training corpus could be classified by an eigenvector decomposition of a special variant of the covariance matrix of the objects. I'm not an expert here, but to me there seems to be a connection to the so called covariance rule (unfortunately I don't have a citation here), which to my knowledge is biologically plausible.
Rogers, T. T, & McClelland, J. L. (2004). Semantic cognition: A parallel distributed processing approach. Cambridge, MA: MIT Press. PDF
Rogers, Timothy T., & McClelland, J. L. (2008). Précis of Semantic Cognition: A Parallel Distributed Processing Approach. Behavioral and Brain Sciences, 31(06), 689. doi:10.1017/S0140525X0800589X PDF