A transformer has a very limited bandwidth.
It has a low frequency limit given by the primary inductance, and the V.s product of the core.
It has a high frequency limit given by the leakage inductance, and the self-capacitance.
As a result of this limit, there are many different transformers available, with different core sizes and permeabilities, and different winding turns, operating over different bandwidths. Most of the wider band RF ones are designed to operate at the very low impedance level of 50 ohms, and are 1:1. Bandwidth rapidly shrinks as transformatio ratios depart from 1:1. You can get low frequency transformers that cover several dacades, for microphones and for 100v speaker lines, but they tend to be big and heavy.
When you pick up an oscilloscope probe in anger (a) you don't want it to load the circuit so much that it changes operation and (b) you're often debugging, so you want to see anything that's going on that shouldn't be there.
(a) The typical impedance of useful low frequency probes, 1Mohm, 10Mohm, with a few pF to 30pF shunt C is way higher than you can get with transformers. Even at RF and microwave where you need lower C but can tolerate higher loading, so you might use 500ohms to 10kohms in series with a 50ohm scope input coax, these impedances are way higher than transformers that will operate there.
(b) With a resistive divider scope probe, you can see reasonably flat from DC to tens, or even hundreds of MHz. With one connection, you can see unexpected DC shifts, or instability at 100MHz, either of which may be a clue as to why your DC regulator, audio amplifier, 10MHz clock, or whatever, are not working properly.
For very specific applications, let's say monitoring ethernet, then the appropriate sized transformer can make a useful probe. However it will only tell you what's happening in the bandwidth you think you want to know about.
