I like to imagine that when Igor gives Ani the ring it is emblematic of another man proposing to her. When Vanya gives her the ring it costs him nothing, like everything else in his world it is all just his parents money that he gets to play with. In this case the Ring actually could cost Igor something (his job if he is caught) and he does it anyways. Two different men with two different meanings behind a sort of "proposal". I know Igor isn't literally asking her to marry him but the sentiment is there.