There are relatively few minerals that streak green or greenish, including copper carbonate rich rocks such as malachite (which I think can be ruled out). Mineral colour itself is not necessarily related to streak colour. Peridot and rocks rich in it are typically green, but it streaks colourless or white, as for example do epidote and olivine. They’re all significantly harder than 5 on the Mohs scale though. Pyroxenes can also streak greenish, with hardness from around 5.5 upwards.
Although the terms are loosely used interchangeably, strictly speaking, serpentine refers to a mineral group of which several will streak green or greenish, and rocks rich in those minerals are referred to as serpentinites. Serpentine minerals generally range in hardness from between 3 to 6 on the Mohs scale. Chlorite often occurs with serpentine group minerals and also streaks green or greenish, but is soft at around 2 to 2.5 on the Mohs scale.
So, yes, your streak colour and hardness are potentially consistent with a serpentinite or something pyroxene rich, but those may not be the only possibilities. Mica minerals such as phlogopite can occur within serpentinite rocks and mica frequently occurs with pyroxenes. Mica doesn’t generally occur with peridot, except as tiny inclusions within the crystals themselves.