The binary asteroid 288P/(300163) is unusual both for its combination of wide-separation and high mass ratio and for its comet-like activity. It is not currently known whether there is a causal connection between the activity and the unusual orbit or if instead the activity helped to overcome a strong detection bias against such sub-arcsecond systems. We aim to find observational constraints discriminating between possible formation scenarios and to characterise the physical properties of the system components. We measured the component separation and brightness using point spread function fitting to high-resolution Hubble Space Telescope/Wide Field Camera 3 images from 25 epochs between 2011 and 2020. We constrained component sizes and shapes from the photometry, and we fitted a Keplerian orbit to the separation as a function of time. Approximating the components A and B as prolate spheroids with semi-axis lengths a$<$b and assuming a geometric albedo of 0.07, we find $a_A leq$ 0.6 km, $b_A geq$ 1.4 km, $a_B leq$ 0.5 km, and $b_B geq$ 0.8 km. We find indications that the dust production may have concentrated around B and that the mutual orbital period may have changed by 1-2 days during the 2016 perihelion passage. Orbit solutions have semi-major axes in the range of (105-109) km, eccentricities between 0.41 and 0.51, and periods of (117.3-117.5) days pre-perihelion and (118.5-119.5) days post-perihelion, corresponding to system masses in the range of (6.67-7.23) $times$ 10$^{12}$ kg. The mutual and heliocentric orbit planes are roughly aligned. Based on the orbit alignment, we infer that spin-up of the precursor by the YORP effect led to the formation of the binary system. We disfavour (but cannot exclude) a scenario of very recent formation where activity was directly triggered by the break-up, because our data support a scenario with a single active component.
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