Is it permissible for me to keep a sum of money that the buyer mistakenly paid me a second time, or is that considered theft?

It is not permissible to take this money because it falls under the category of unjustly consuming people's wealth, as Allah Almighty says: "And do not consume your properties among yourselves unjustly," and as the Prophet, peace and blessings be upon him, said: "The wealth of a Muslim is not lawful except with his willing consent." And whoever gave you money due to their heedlessness, their soul would not be content with it. The matter becomes even more severe if the purchase is for a government company; in this case, it is not permissible to take it even if the giver's soul were content, because one does not have the right to dispose of public funds except in a way that serves the public interest.