Did people back then build supercomputers out of multiple smaller ones?
For one, multi processor systems have been around since the late 1960s. For example the IBM /370-168 could be ordered with a second processor unit. It was a for all computing parts symmetric, except only the first CPU could hand over I/O jobs to the I/O processor.
IBMs 3033 of 1978 did away with this limitation by having two CPUs and two IO processors where each CPU could start each IOU and each IOU could interrupt each CPU. The 3033 could also come in a tight coupled dual system via a high speed connection – maybe think of it like two computers with dual core CPUs connected via a multi lane PCIe link.
Build supercomputers out of >1 smaller ones
Since there where no 'smaller ones', there was no real gain in doing so. Today's parallel systems rely on the fact that sufficient fast single chip CPUs are available at extremely low prices. Back then super computer (or basically any fast machine) CPUs where build from dedicated chips, developed especially for that machine. Gaining more throughput could be achieved by adding special function units and other optimizations in a way more effective way than piling up CPUs.
Also, it's important to keep in mind that parallel CPUs only speed up very special classes of jobs. Again it was more effective to construct a CPU in away to gain functional parallelism than adding general purpose CPUs. The Basic principle of CDC and Cray machines revolve about the idea of a set of (virtual) I/O processors feeding function units in a way to keep them as busy as possible.
Time range approximately 76–82 (I accept exceptions, especially earlier than this time range, but not more than approximately 90)
We could as well start with the ILLIAC IV of 1973 (working) with 64 parallel FPU units. Ok, given, they where special purpose, so the next step might be the Connection Machine 1 of 1985. Here up to 65Ki independent CPUs, each with its own memory and communication where tied up in one supercomputer.
There have been other attempts at massively parallel systems in the 1970s, but in all cases it was cheaper to get the same or even higher computing performance by improving existing architectures. Especially by not having the need to rewrite application.
An early example of using a massive parallel system with (rather) standard processors was the BBN Pluribus networking system in 1978 build up from 128 PDP-11 (alike) nodes. Except, its job was message switching, not computing – and nowhere near in computing power what supercomputers of the same time delivered. Communication was done via a shared memory architecture.
The follow-up BBN Butterfly with up to 512 CPUs of 1985 comes closer by using of the shelf 68k CPUs, but still it was a switching system, not a supercomputer.
In 1983/84 the first of INMOS' Transputer CPUs, especially made for massive parallel systems (only 4D connections max), and companies like Meiko and Parsytec soon constructed machines using them.
Eventually the first machine that is quite similar to some architectures in use today would be the Intel iPSC/1 of 1985, based on 32–128 compute modules, each with an 80286/87, 0.5 MiB of RAM and 8 Ethernet ports for communication, wired up as a 5, 6 or 7D cube, depending on the number of nodes. Later, also 80386 nodes with Weitek (!) FPUs where available.
nCube did build a quite similar system, except being scaleable up to 1024 CPUs. They are btw a great example that the concept of massive parallel systems where mostly a solution without a need. After first attempts on scientific computing they soon turned into supplying massively parallel video servers, like for Hotels and alike.
Eventually the first system true to today's image of a massive parallel computer build from standard CPU nodes might be the SUPRENUM-1 of 1990. This is not only true about the hardware, but even more so for software, as they where the first to supply a massively parallel capable FORTRAN compiler as part of the project, as well as many research about parallel algorithms and how to analyze and structure application for parallel computing. Many of their tools and libraries can be found are the core of today's software.
Again with special CPUs, the Cray 3/SSS was delivered with up to 256k CPUs in 1995 – shortly before the original Cray Corp went belly up.
I'm referring mainly to hobbyists, although maybe very very small companies should be included.
Why should any hobbyist do so? What would have been the need for that. I mean, beside the extreme cost. As a hobbyist one was happy to acquire a single computer. Buying a dozen or more is way outside a hobbyists reach. Which by the way, it still is. A somewhat capable CPU board is still several hundred bucks, and even buying a dozen of them will not make it as fast as a top end Ryzen board will be – at lower cost! And buying a dozen Ryzen boards each for like 4,000+ USD (CPU + sufficient fast RAM) catapults the project past the 50 grand mark – definitely not hobbyist territory.
Combining a bunch of cheap boards – like a dozen RPi's – doesn't make a supercomputer, as their combined computing power will be dwarfed even by any of today's Office PC class CPU with integrated GPU.
The same is true when going back.
And then in 1980? A hobbyist reachable CPU at that time was something like a 6502 or Z80 – even 100 of them wouldn't even come close to an average mainframe of the same time, even less a supercomputer.
Could they just have connected multiple smaller/cheaper machines, via network cables, to build a supercomputer.
Sure they could, but like described before it wouldn't make a supercomputer, just a bunch of small cheap machines connected. In 1990 this premise would have been like taking several 8–10 MHz 80286/287. A 10 MHz 80286 reaches about 0.15 Whetstones. A top of the line 486 in 1990 would be a 50 MHz one delivering about 20–25 Whetstones – that's about 120–150 times what the 286/287 does. So to even replace one up to day 486 computer with at that time cheap computers would need >150 of them. Besides that already the cabling and LAN cards will cost more, I do not want to pay the electricity bill for that setup.
So while 150 80286 would barely make up a 80486, a real supercomputer of 1990, like a Fujitsu 2400 or a NEC 3/11 would play in a region of 2000 Whetstones or above. Or like again 100 times faster. And all that without taking the communication overhead into account such standard of the shelf solutions have compared to the highly optimized supercomputers of that time. There is a reason that it took another 15+ years until standard CPUs did really reach the top of super computer power – and even that only by using extreme amounts of them in very tight coupling – not just PC-Boards connected via some network.
Or could they have removed processors from multiple machines to make a supercomputer (parallel-processing )
And put them on what boards?
Parallel-Processing/Parallel-computing back then by using >1 smaller machines, from my limited knowledge it would not have been possible/too difficult.
Sure it was possible – see the above examples about serious machines. But for a hobbyist it would have been as impossible back then, as it is today.
EDIT – I'm more interested in this being done using affordable common home-computers, and in fact the cheapest home-computers, but almost any examples within the parameters of this question are ok.
Bottom line, there are none, as this question is based on an assumption that just doesn't add up. In all times since we have computers, and even more so since the microprocessor, there was never a use case to combine low power CPUs for more computing power, as using higher power versions of the same type will beat such a combination by at least a magnitude, while doing so at a fraction of the cost.
There is a reason even today's supercomputers are never built of middle of the road CPUs, but always the top end.