Context. High-resolution spectroscopic surveys of stars indicate that the Milky Way thin and thick discs follow different paths in the chemical space defined by [α/Fe] vs. [Fe/H], possibly suggesting different formation mechanisms for each of these structures. Aims. We investigate, using the Gaia-ESO Survey internal Data-Release 2, the properties of the double sequence of the Milky Way discs, which are defined chemically as the high-α and low-α populations. We discuss their compatibility with discs defined by other means, such as metallicity, kinematics, or positions. Methods. This investigation uses two different approaches: in velocity space, for stars located in the extended solar neighbourhood; and, in chemical space, for stars at different ranges of Galactocentric radii and heights from the Galactic mid-plane. The separation we find in velocity space allows us to investigate, using a novel approach, the extent of metallicity of each of the two chemical sequences, without making any assumption about the shape of their metallicity distribution functions. Then, using the separation in chemical space, adopting the magnesium abundance as a tracer of the α-elements, we characterise the spatial variation of the slopes of the [α/Fe]−[Fe/H] sequences for the thick and thin discs and the way in which the relative proportions of the two discs change across the Galaxy. Results. We find that the thick disc, defined as the stars tracing the high-α sequence, extends up to super-solar metallicities ([Fe/H] ≈ +0.2 dex), and the thin disc, defined as the stars tracing the low-α sequence, extends at least down to [Fe/H] ≈ −0.8 dex, with hints pointing towards even lower values. Radial and vertical gradients in α-abundances are found for the thin disc, with mild spatial variations in its [α/Fe]−[Fe/H] paths, whereas for the thick disc we do not detect any spatial variations of this kind. This is in agreement with results obtained recently from other high-resolution spectroscopic surveys. Conclusions. The small variations in the spatial [α/Fe]−[Fe/H] paths of the thin disc do not allow us to distinguish between formation models of this structure. On the other hand, the lack of radial gradients and [α/Fe]−[Fe/H] variations for the thick disc indicate that the mechanism responsible for the mixing of metals in the young Galaxy (e.g. radial stellar migration or turbulent gaseous disc) was more efficient before the (present) thin disc started forming.