"Bias current" is incorrect in this context; it isn't an op-amp input!
WRT this: There are quite a few diffprobe designs floating around that use opamps configured as differential amplifiers. Quite a few on this website, in fact.
To be more accurate, diff probe is actually nothing more than instrumentation amplifier. A special version that is either really fast or has really robust inputs with large common voltage range.
And I guess you could say they have input current that you could call bias, but it is not called so. With probes, it is simply input impedance, both differential (between the inputs) and single ended (towards "ground").
And they all have third connection for common reference potential.
High speed low voltage probes come with "ground" terminal because of low common mode range and to maximise CMRR and minimise various spurs in signal...
Connection to the ground will be short and local to DUT.
For a high voltage probe that is not so critical, so their third terminal ("ground", I actually like this quoted term for common reference potential) is directly connected to scope grounding.
Any common voltage on inputs gets suppressed by CMRR of the probe. That is why they don't usually have third connection, except some do, and a warning it has to be connected to protective ground or some common reference plane in SUT if your scope input is floating (is isolated from common frame of reference). That would happen if you (not recommended) float your scope, or if you have proper isolated channel scope, in which case you mostly don't need diff probe. Unless you want to use diff probe anyways, because it is more robust, or you want to repeat previous measurement so you want to compare with the same circuit loading (same probe).
One not very much discussed point in having a diff probe is mostly overlooked. People speak about reference planes and grounds, dangers and such.
To me, best explanation why you need diff probes, and their most useful characteristics is this:
With classic grounded (passive or active) probe, you take GND wire from the probe and connect it to GND of your circuit. And then you probe various SINGLE points in circuits. So you are observing a signal at certain net in a circuit. So if you imagine stack of batteries, you can measure voltage on first battery, and then voltage on second battery (that will be in series with first one) etc.
If you wanted check a voltage of every individual cell (you want to check if cells are good), you cannot measure voltage ACROSS every cell, like you can with multimeter. Connecting grounded probe across third cell will connect scope ground in such a way that it will short first two cells and kill your scope if there is enough current in them.
Diff probes will allow you to do just that. With them you can measure across the components. Between two arbitrary points, not only potential on that net in regards to ground.
Like you do with multimeter, and that makes them very useful when trying to understand what is going on in a circuit.
Of course, with usual disclaimer that they have input impedance (complex one) that is usually lower that other probes. So they can interfere with high impedance circuits.
Like any probe. connecting them to circuit can change circuit behaviour.. So you have to understand both probe spec and circuit you're measuring.
Probe is a part of the circuit when connected.
* CMRR is common mode rejection ratio, for those that don't know.
SUT- system under test