How did the Indo-Greek kingdom influence Indian society/culture/history?
The short answer is "Gandhara statuary and the
Milindapanha".
Art historians claim that there is a very definite change in the sorts of poses and details emphasized by sculptors in classical subcontinental art following the Indo-Greeks, with definite connections to Greek types already in evidence further west. For instance, it is argued that the first depiction of the Buddha as a
human figure as opposed to a
symbol - leading to the extremely famous types of seated Buddhas known from all over the Far East - is due to Greco-Buddhist artwork. This claim is generally dismissed by Indian nationalists in an effort to prevent anything important in 'Indian' history from being the work of 'foreigners', and since it's art history it's kind of hard to prove them wrong.
The
Milindapanha, or Questions of Menandros, is a work in which the Indohellenic king Menandros (I
Soter) engages a Buddhist sage named Nagasena in philosophical discourse. As part of the Pali canon, it's one of the most highly regarded works of classical Indian prose. This one's hard for Indian nationalists to argue, because Milinda is universally identified as Menandros and more or less everybody agrees that it's a pretty good work of philosophy. So they claim that, at that point, the Indohellenic kings were, in fact, "Indian" and not "Greek" at all, which is more than a little ridiculous.
That's for culture, and it's the short version. Society is a bit tougher, because we know virtually nothing about the societies of the Indohellenic kingdoms. Pretty much everything is extrapolation or out-and-out guesswork with very shaky foundations. W. W. Tarn once tried to come up with a fairly comprehensive description of Indohellenic society, but people have been chipping away at it for the last eighty years, so.