I am thinking on analog circuits, whose circuit design could be read (or, at least, significantly altered) from software. Do they already exist?
There are many chips that can be configured by choices made in software, which then have a perfectly analogue behaviour that gets altered.
Blocks that work like that go from Analogue Switch arrays and/or Multiplexers all the way past Programmable-Gain amplifiers through to complete integrated chips that have a combination of blocks, such as programmable gain, a whole bunch of analogue switches, analogue mixers and multiplexers and all such. Analog Devices are kings of doing things like that.
As an example a programmable gain amplifier may have only a discrete number of gains, 4, 16 or 32, or even 256 for a very high-end, luxury one, but it will gain any voltage on the input by that number, any real voltage that can be made by something, so they are truly analogue.
I believe there are even some actual FPGA-type devices, much like DSPICs, that include some of those kinds of blocks into the programmable gate array along with DACs (Digi->Analogue) and ADCs (Analogue->Digi), but no types spring to mind, so I could just be misremembering.
For sure many DSP System on Chip (SoC) devices made for old-school analogue video and/or multi-track audio manipulation and conversion would be a mix of such analogue trickery before the signal would be turned digital and processed further as a digital signal. Which then in some cases would be turned back to analogue and possibly again multiplexed or modulated into some kind of antenna signal.
No modern-to-future-day applications for such a complex SoC other than very specific Audio tools or metrology spring to mind right now, but it's not unlikely they still exist, since no one person will have heard of everything.
So, in more ways than one, the answer seems to be yes, they exist. Though of course not exactly like an FPGA, because a digital beast can never be fully compared to an analogue one.
Yes, it's called Field-programmable Analog Array (FPAA):
They are exactly the analog version of the FPGAs. There are many different subtypes (current or voltage driven, discrete or continous time). They are built up from CABs (configurable analogous blocks), and from interconnects between them.