Short Answer
Yes, by most conventional definitions, the human brain is an analog computer with the caveat that there are a range of definitions for 'computer', so it's best to use an abstract one.
Long Answer
Sufficiently Generalized Definition of Computer
Before we tackle the answer in question, let us observe that according to prototype theory which springs from Wittgenstein's notion of family resemblance, it would be silly to bicker over details of definitions of computers that stem from necessary and sufficient definitions for more concrete examples. Computers can be embedded, general purpose, digital, analog, distributed, biological, electromechanical, or even biochemical, as in computations that are performed with nucleic acids. Abstracting from different sorts of computers leads to an idealized definition of a computer that is frequently seen in introductory textbooks on information systems and information technology:
A computer is any system which embodies the information processing cycle.
This is vague and useful as one can get, and very much lends itself to setting up a short ontological razor with the four phases of the information processing cycle: input, storage, processing, and output.
The Watch, an Abacus, and a Cup of Coffee
Given the definition above, what is important is that a system embody all four phases of the information processing system. Let's think about two devices that commonly held to be computers: the watch and the abacus. In the case of a watch, even a mechanical one, all four phases of the information processing cycle are present. First, you have to enter a time. The point of a wrist watch is to keep a number which represents the time of day synchronized with the world. So, a user enters in a reasonable approximation of the current time. This is input, and on a mechanical watch, is often accomplished with a small dial. Now that the time has been entered, it will change at a predetermined rate. If the gears are relatively continuous, then the watch is an analog computer, and if the gears allow for a punctuated smallest transaction, say the transition from 1s discretely to the next, the system is digital. This is the processing aspect of an information system. Of course, the time that is stored/represented by the watch is within the gear structure itself, which is the way the watch stores the current time. Now, that value is visible to the user traditionally with a hour, minute, and second hand, so the face of the watch ultimately allows for output. It might be possible, for example, for the internal mechanism to function correctly and the hands to be damaged. A watch therefore automates a number system to track the passage of time. Time and numbers are abstractions, so ultimately they are forms of information.
An abacus, which is often touted as an example of a computer, is not a computer under this definition. Why? Let's follow the same train of logic. A user certainly can configure an abacus or quipu to represent a number. This would be inputting a number. And at a glance the user could then, days later, take a look at that representation of the number in the physical medium demonstrating that the number has been stored and is capable of output. But where an abacus does not fulfill our definition of the computer is by its inability to automate information processing by having a processing phase. Traditional abaci simply do not transform the informational representation in the same way a watch or a microprocessor is capable of. One often hears the argument that the beads slide, for instance on a frame, and that is processing, but that processing really is not the abacus, but the user of the abacus. In this way, an abacus is more like a pencil and a pad of paper. You can write a number on it, and days later retrieve that number long after you have forgotten it, but unlike a mechanical watch or a digital CPU, there will be no change of state without the user. An abacus no more computes than does a math problem scrawled on a sheet of paper. Thus, neither are computers.
Now, along this continuum we can refer to the coffee cup, which is also a physical medium, does have input and output, and does allow for a change in state (let's say by the cooling of hot coffee). Does this satisfy our four phases? We can add coffee. It remains there for days more or less. We can drink coffee from it. And we might be able to use it to mix creme, so there are chemical and thermodynamic processes at work. This is input, output, processing and storage, right? Yes, however, what is missing from the system is an abstraction called information. In the physical/mental dichotomy, we generalize phenomena between those like numbers and those like molecules, and unless somehow the coffee can be made to do work on information, it is simply not a computer. Yes, but can't we measure the properties of the coffee or the cup? Yes, but again, is the system constructed and functional as an information processing cycle, and the answer is generally no. I've never heard of someone pouring a cup of coffee to do calculations, which places it in the non-computer sort of processing.
But aren't all computers physical processes too? Yes. Of course. Information does not exist without a medium. Information is part of a system generally defined with the SMCR model. For information to exist, you have to have a generator and sender of information, that information must be physically embodied in a medium, and there needs to be a receiving agent. Frequently, noise is introduced into the message through the medium, and communication processes and information theory are very interested in such processes and the mathematical truths that characterize these systems. In fact, the interconnection of information processing machines is now so common that computers and the systems for sharing information are often collectively referred to as ICT, information and communications systems.
The Human Body and Brain and Computers
Is the brain essentially a computer? The short answer is yes. John von Neumann in his classic The Computer and Brain actually does an analysis because the biological brain and the electromechanical digital computers have much in common. Now, clearly there are many differences, and von Neumann details those differences. But it is irrefutable that both ENIAC and the human CNS embody the information processing cycle. Both systems can accept a pair of numbers, recall those numbers after an elapsing of time, add those numbers, and then pass that sum outside of their respective systems. This is the foundation for the idea that computers are capable of some degree of intelligence; but mightn't one object that computers are made of inert physical matter and are programmed whereas humans are alive and have free will? That is of course a knee-jerk reaction, and doesn't hold much water upon reflection and smacks of vitalism. It was once thought that organic and inorganic chemical compounds were somehow different stuffs, but that was shown to be false. Descartes famously argued that the lack of language use meant that animals were mere automatons. (See "Quotations from Descartes on Animals as Automata" (PhilSE).) Many still believe that computers are mere bit-twiddlers whereas people understand meaning. (I push back on that in "Computers, Artificial Intelligence, and Epistemology" (PhilSE).) While the nature of AI and AGI are long, complicated subjects, the short order of it is that what separates the intelligent from the unintelligent is very much open to debate, and tests like the Turing Test or thought problems like the Chinese Room are very much philosophically difficult characterizations, both to claim and to refute with no clear consensus emerging among even the brightest thinkers.
Summary
Yes, the brain functions like a massively parallel analog computer (though neuron firing patterns are clearly both digital and analog). Yes, digital systems are essentially punctuated analog systems. Yes, 0s and 1s represented with voltages are created from square waves that could readily be sinusoidal at the whim of the architect. No, the brain does not embody an architecture remotely similar to the Harvard or Von Neumann architectures. Yes, they both use electricity, but one is clearly wetware. No, it doesn't matter what the substrate for computation is. Yes, both can clearly process numbers, language, and visual images. No, they do not do it in the same way. Yes, one is programmable to a much greater extent than the other. But that doesn't mean the other can't be given and expected to follow instructions. Yes, the human brain evolved to process information for survival value. No, current artificial computers do not have a mind in the sense that great apes do. But the continuum between worm CNSes and human CNSes is one of degree. But, to deny that human brain is a computer is be largely to affirm a lack of knowledge on what makes a system a computer to begin with.
