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I have calibrated pH sensors in the lab on several occasions and have used standards of 4.0, 7.0 and 10.0 usually. Recently I received a sensor and was tasked to calibrate it, but it requested standards of 6.86 and 9.18 in addition to the 4.0 — with no ability to change this. I see that you can buy these standards but they are kinda tricky to come by.
My question is: Why these specific numbers, and who uses these? (as I have never seen them before)
The pH 6.86 and 9.18 values come from NIST standard buffer solutions for pH calibration, as described in, for example, NBS special publication 260-53 (1988) (pdf available here).
It appears they chose those values because they are close to the desired values of 7 and 9, but are also easy to prepare. For example, the 6.86 results from combining equal concentrations of $\ce{KH2PO4}$ and $\ce{Na2HPO4}$ (see Table 1 of the above-referenced document), whereas the 9.18 buffer is a conveniently round number concentration (0.01 M) of borax.
The reason that you might be required to use these standards is to ensure that your calibration meets NIST standards.
(For those outside of the USA, NIST is the United States National Institute of Standards and Technology, formerly the National Bureau of Standards, responsible for national-level standardization of measurements.)
Those strange $\text{pH}$ values belong to the primary $\mathrm{pH}$ buffer standard solutions, that have their $\ce{pH}$ value given by the definition of the practical $\mathrm{pH}$ scale. It means it is valid worldwide in context of metrology, not being limited to NIST. There is implied the standard temperature for electrochemical measurements $\pu{25 ^\circ C}$.
The round values belong to secondary standard $\mathrm{pH}$ buffers, that are often tabelized for a suitable $\ce{pH}$ range. During their development, their $\mathrm{pH}$ values are adjusted using a pH-meter calibrated by primary standard buffers.
See e.g. Primary pH standards on iccontrols.com :
4.005 pH, 0.05 mol/kg potassium hydrogen phthalate
6.865 pH, 0.025 mol/kg disodium hydrogen phosphate + 0.025 mol/kg potassium dihydrogen phosphate
7.413 pH, 0.03043 mol/kg disodium hydrogen phosphate + 0.008695 mol/kg potassium dihydrogen phosphate
9.180 pH, 0.01 mol/kg disodium tetraborate
10.012 pH, 0.025 mol/kg sodium hydrogen carbonate + 0.025 mol/kg sodium carbonate
Note that disodium hydrogen phosphate comes as $\ce{Na2HPO4}$ (hygroscopic), $\ce{Na2HPO4 . 2 H2O}$, $\ce{Na2HPO4 . 7 H2O}$ and $\ce{Na2HPO4 . 12 H2O}$(the most usual, AFAIK). The crystal water has to be reflected in used masses of the phosphate and water, to keep the phosphate molar amount and the total water mass.
