Not to be divisive, but...not completely. I would argue that anarchism is more comprehensive in its desire to eliminate hierarchy and authority than socialism. Anarchists are socialists, but most anarchists want to go further than socialism, and we all want to do it in a way that respects means-ends unity, i.e. the means we use to achieve our ends should resemble those ends.
For this reason, even though anarchists and other socialists both want to abolish capitalism and put workers in control of the means of production...we have wildly different visions for what that looks like. We share a common language primarily, and many common values (e.g. egalitarianism, anti-racism, anti-fascism, feminism, anti-LGBTphobia, internationalism, anti-imperialism, anti-capitalism), but the theoretical and practical divide between anarchism and Marxism is fundamental.
That being said, non-anarchist socialists are absolutely leftists. No question. We do share enough commonality to share that label. But we do not have exactly the same goals.