It seems pretty straightforward to me that with a European dominated government deciding the border of Kazakhstan that they would choose to define Kazakhstan as ending at the Ural mountains and river. In spite of that why is it that the border of Kazakhstan is a fair bit west of the Ural river? Is there a site significant to Kazakhs as a people in that region? Was it something really dumb like a cartography error? The alternate history community even has a common expression related to the arbitrariness of the border https://www.alternatehistory.com/wiki/doku.php?id=alternate_history:that_damned_kazakh_border so how did it originate?
The Ural River in fact was a border used by the Russian Empire: the cities of Uralsk and Atyrau were originally known as Yaitsk and Gur'yev (also as "Lower Yaitsk") respectively, and were established as Cossack fortifications in the late 16th and early 17th centuries. Uralsk is in many respects a very Russian city, and is older than St. Petersburg. Pugachev's 1774 rebellion actually started among the Cossacks in Yaitsk/Uralsk.
Anyway, although there was an imperial border on the Ural River, it was never a very hard border, and is better understood as a border zone. Kazakh nomads were frequently allowed to cross the river to graze their herds on the Western side.
In 1801, because of increasing pressure on grazing lands east of the river, 7,500 families from the Kazakh Small Horde were allowed to cross to the western side and settle there as the "Inner Horde" (also known as Bukey's Horde for the Khan who led them). This community, like the better-established Kazakh Hordes at the time (there are three hordes Junior/Small, Middle and Senior/Great), was internally self-governing, but effectively a Russian imperial protectorate until the 1840s, when the position of Khan was abolished and the area governed as part of the Governor-Generalship of the Steppe.
From then until 1991, it's worth keeping in mind that the borders were internal administrative borders, not international ones. When the Kirgiz Autonomous Soviet Socialist Republic was combined with ethnic Kazakh portions of Turkestan and renamed the Kazakh SSR in 1925 the intention of Soviet authorities was to create an autonomous Republic (elevated to full SSR status in 1936) that would serve as a titular homeland for Kazakh people, as per Soviet nationalities policy. By that logic it made sense to include areas governed under the Steppe Governate, as well as areas near Astrakhan, that were mostly inhabited by ethnic Kazakh communities.
But as internal Soviet borders, Soviet authorities never treated these as the international borders that they have now become. For instance, the Trans-Siberian railway crosses through Kazakhstan at Petropavlovsk before continuing back into Russia. Likewise, the train line that connects Uralsk to the Trans-Aral railway line leaves Kazakhstan, passes through Russia, and re-enters Kazakhstan before connecting to the main line.
TL:DR the border isn't arbitrary but reflects a couple centuries' worth of Kazakh settlement west of the Ural River. Nevertheless the border was until 1991 an internal administrative border in any case.
Source:
Martha Brill Olcott. The Kazakhs