The British prime minister and cabinet are all elected. They are elected MPs like any other, who are selected by their party to perform additional duties. That is not at all the case with the members of the European Commission.
1. When we're talking about this, this is generally about an election to the executive office the person holds, not the legislature. Plenty of countries have cabinet positions that are specifically not drawn from the legislature as the result of separation of powers. After all, part of the reason for an election to executive office is the control of the executive by the legislature.
2. British cabinet ministers can also come from the House of Lords and not just from the House of Commons. This is rare in modern times, though the Leader of the House of Lords, a cabinet position, always is a member of the Lords.
