Activation functions in NN map to action potentials/postsynaptic potentials, weights map to synaptic strengths/connection patterns between neurons, dropout to synaptic pruning/apoptosis, convolutional layers from the receptive fields of our visual cortex.
Certainly our brains are more complicated. Temporal spike timing, dendritic signals, synaptic plasticity, compute of synapse themselves, lots more I can't recall, none of these concepts are currently modelled in our NN architectures.
NNs start from the simple premise of interconnected neurons. As the field develops and we find what works or what doesn't, perhaps we will take more bits of inspiration from the brain.
A Perceptron model is just a dot product: one multiply+add, summed to one output scalar value. Modern nomenclature might consider Perceptrons to just be "one channel of a single layer," or like "one fully-connected layer with a single output." In fact, that's where the old-fashioned name for neural networks came from: "Multi-layer Perceptrons (MLPs)" are a bunch of these arranged on top of each other. The older 1943 work showed that in some sense lots of possible arrangements are roughly equivalent to each other so it made sense for Rosenblatt and friends to start from the simplest model with the understanding that the field would build from there. (Our feedforward model ancestry is just one branch of the tree rooted in this work; there's a completely separate geneology for other possible arrangements of modeling components, like recurrent Boltzmann machines that simply haven't been so well studied/understood... :-)
The generality of Perceptrons made room for a lot of interpretive flexibility. Right from the start, news reporters saw the theory, attended academic conferences, and started writing about it. The NYTimes had an article in 1958 that spoke of the Perceptron as "the embryo of an electronic computer that [the Navy] expects will be able to walk, talk, see, write, reproduce itself and be conscious of its existence."
In fact there was a very famous kerfuffle between Rosenblatt (Perceptron authors) and Minsky & Papert (another AI researchers). The latter two published a spicy book in 1969 saying that a single Perceptron wasn't general enough because linear models can't even learn an XOR function. Everyone who read this book misinterpreted it as a takedown of Perceptrons; this book inadvertantly dried up interest, choked funding across the board, and ultimately caused the first AI winter which didn't get resolved until well into the '80s when these ideas started to be revitalized. You can read about the spilled tea here, it's quite fascinating: https://doi.org/10.1177%2F030631296026003005
Choosing positioning like "NNs are similar to the brain" or even calling the field "machine learning" makes it harder to speak objectively about the research we perform, makes it hard to understand the sort of explanatory power and limitations of the models we fit, and makes it harder for users to understand how the system works. It's like how quantum mechanics researchers have to deal with readers who misunderstand what linear observable operators are and conflate it with their own ideas about conscious observers or use it as evidence of God's presence or whatever.