The transformer is there mainly to decouple DC common mode signals, as has been explained in other answers, and it also provides the electrical isolation needed in practical data networks (as opposed to running the entire network on your bench, all powered from the same extension strip, for example). The transformer manufacturer will state which IEEE 802.3 sub-standard it complies with. Usually, transformers for 1GBASE-T can work all the way down to 10MBASE-T speeds, and it seems that 10GBASE-T transformers retain similar performance, but it doesn't guarantee outright that a 10GBASE-T transformer will work with e.g. a 100BASE-T PHY chip, since some key parameters have changed.
But it certainly is at least theoretically possible to e.g. design a 10M/100M/1G/10G BASE-T digital signal processing chain in an FPGA (or a modern GPU, or an ASIC), attach it to an A/D-D/A front-end running at 800MS/S and using 10GBASE-T magnetics, and have a solution compliant with speeds from 10M to 10G - over 3 orders of magnitude. But in reality nobody designs such "wideband" digital processing chains, since it'd be wasted effort as 10G ports are so expensive that plugging anything slower into them is a total waste of money. Typically 10G PHY will be gigabit speeds only, often not even 1G but higher: 2.5G, 5G and 10G. Some are 10G only!
Twisted-pair Ethernet at speeds of 1GBit and beyond work like analog phone systems: a single pair is used to transmit data in both directions at once. Analog phone systems have a single differential "loop" connection: it's not that one wire transmits and another receives. It's a single closed circuit that works in two directions as the same time. It works on the principle that the sender has all the information needed to suppress its own signal from what it receives from the loop.
Each end of the connection is both a transmitter and a receiver. The transmitted signal is superimposed on the signal already existing on the circuit. The receiver then measures this summed signal, and subtracts its own transmitted signal from it. What's left is what the other transmitters on the link have been transmitting. Since such links are usually point-to-point, once you subtract your own transmission from the received signal, what's left is the other transmitted signal from the far end of the line.
The subtraction of the transmitted signal from the overall received signal has to be done whether we're talking about gigabit Ethernet or two-wire phone lines. On phone lines, the signals are slow enough that a passive device called a hybrid - essentially a transformer with a termination - can perform adequately. Such hybrids are present in analog phones and other simple phone equipment. The transformer-based approach doesn't work very well when e.g. geostationary delays are present, and is insufficient for high-speed modems, and thus even modems that work on phone lines have to implement a "hybrid" in the digital domain. They still use transformer for isolation, and may connect it in a hybrid fashion, but the performance of that transformer is inadequate by itself.
Once it became cheaper, in aggregate, to work on digital signals using chips that weigh micrograms rather than making and shipping chunks of iron and copper known as hybrid transformers (that individually weigh more than all the semiconductor chips in your PC, combined), the function of a hybrid in voice phone networks has been implemented by digital signal processing. As an example: a lowly ARM M0 microcontroller has enough computational capacity to implement an adaptive telephone hybrid in software, with excellent performance, and its manufacture uses much fewer resources than the manufacture of a telephone hybrid transformer. There are benefits to modern technology being lightweight that go way beyond mere convenience :) A lighter, simpler transformer is still there to isolate the phone line with its voltage transients from the relatively sensitive rest of the circuit.
As the data rates increase, so does the computational capacity needed to perform this separation of transmitted and received data, and of adapting to the imperfections of the cable plant (e.g. each connector/jack or kink causes echoes!). So, by the time you look at a 10GBASE-T PHY chip, it has about as many transistors as the Pentium III (Katmai): ~10 million. Of course only a part of this transistor budget is relegated to near- and far-echo cancellation and equalization, but it's still not an insignificant chunk. Guesstimating from block diagrams provided by e.g. Broadcom, a good 1/10th-1/3rd of the chip deals with just that (I'm not dealing with such chip design and don't have any rules of thumb handy to provide a narrower estimate). The digital filters used for echo cancellation and path equalization have lengths on the order of 100-1000T (taps). That means that the PHY has to do a few thousand 10-12bit multiplies and additions per each sampling period just to separate the transmitted signals from received signals, and to equalize away the imperfections of the cable system; and there's 800 MSamples/s needed to run 10GBASE-T. So we're talking on the order of 1TMACs/s (that is one Tera MACs/s, or one million million multiplies-and-additions per second).
You could emulate this using a modern GPU. For perspective: an early Intel HD Graphics platform from 2010 could perform maybe 0.03TMAC/s. Intel Iris Graphics introduced in 2013 could handle about 1TMAC/s at half precision - and would just about have the capacity needed to perform the equalization and echo cancellation for a single 10GBASE-T transceiver. And this is just to get "clean" digitized signals that you still have to demodulate, descramble, decode, and error-correct (and do the reverse when transmitting). That's easily another equivalent of a few hundred GFLOPs/s, I imagine, although quite a bit of it is not floating point but parallel bit operations and fast memory lookups.
An aside: you could have more than two transmitters distributed along the line, as long as their signals could be otherwise separated. Satellite positioning systems deal with this: all GPS satellites send at the same frequency, but their signals are each coded differently, so that if you use a code that decodes one satellite's data, all the other satellites's signals get turned into noise that can be separated from useful data. This is called code-division multiplexing. But this was truly an aside, since GPS is one-way-only: the transmitters are the satellites, and the "GPS devices" like the phones are receivers only. Attempting to transmit at GPS frequencies will get you in legal hot water, and these days fairly quickly at that (never mind that it's totally pointless: the satellites aren't listening to you!).
But you could in principle use code-division multiplexing to e.g. drop several 10GBit/s equivalent PHYs on a single "ether" - e.g. use baluns to couple those PHYs to a good old thinnet 75 Ohm coax. Then, let's say each of the PHYs would be limited to transmitting at 10MBit/s. You could have dozens of them, all transmitting at once, using different code parameters, and each one of them could instantly hear all the others and produce the 10MBit/s received stream from any other - in fact, with the resources of a custom 10GBASE-T PHY, it could decode many of those transmissions at once. So, with modern technology, a "good old" 75-Ohm coax 10MBit Ethernet could be switched from TDMA to CodeDMA (not CDMA!!) and allow collision-free network segments having the number of nodes typically present on those segments back-when (from a few to a few dozen). It would not be possible to get a full 10GBit/s bandwidth from just one coax, but I imagine that 1-3GBit/s would be possible... with each network card using computational resources of a thousand Cray-1 machines. Now you all know why they didn't have code division multiplex Ethernet in the 80s - it's quite elementary: Cray-1 had a production run of about 100 units. Had they made about 2000 of them, you could use each 1000 to put together a CodeDMA 10BASE-T node to demonstrate it all. Also - back then the necessary ADCs and DACs were mostly fiction, so the implementation would need to be done using slower converters with intermediate frequency translation, and the digital processing would have to re-translate the sub-bands into the wideband baseband time-series format, and then out of it on the transmit side. But of course, the limiting factor was the poor availability of supercomputers, mind you :) Yes, optical FFT maybe could be leveraged to implement some of those FIR segments. But back in the 80s that was fairly secret stuff :)