Coincidence. Or, more accurately, the expulsion of the Tarquins did not actually occur in 510, the same year as the expulsion of the Pisistratids, and the first magisterial terms did not occur in 509. This is a date that only occurs in Livy, although it was probably also in Fabius Pictor as well. The dates of much of Livy's first book are pretty much nonsense and the Romans knew it. Additionally, Livy attempts often to tie early Romans with Greeks. It is in this vein that Livy mentions a fictional expedition to Delphi by Brutus, as well as the constant parallels between Greek and early Roman battles (see Horatius and the expedition of the 300 Fabii, both obvious parallels with Thermopylae and attempts to outdo them). The date of the expulsion of the Tarquins is pretty much fictional--the kingship must have ended around then, although Etruscan influence in the city goes quite strong well into the 5th Century. The Roman Republic, with its system of magisterial elections by popular assemblies and the purely advisory duties of the senate was recognized as a form of direct democracy, a point that Livy makes elsewhere, such as in the Revolt of the Plebs. By tying it to the Athenian expulsion of the Pisistratids Livy is attempting to make a point.

The beginning of the Roman republic is traditionally 509 B.C. when the last king Tarquinius was expelled and the office of consuls was introduced.

As for Athens, their democratic system was reinstated in 510 B.C. after it was disrupted by a tyrant called Peisistratos.

As you can see, saying that the Roman Republic was founded within a year of the Athenian is a quite a stretch, but this is one possible explanation.