According to its recently quoted annual report, legislation treats it "as a corporation". In what sense is this true and in which is it not true?
Although the word typically refers to a specific type of business entity, "corporation" simply means a group (or "body") of people or other entities that are collectively endowed with a status that treats them as though they were a single legal person that can exist indefinitely.
In common-law jurisdictions like the U.K., this includes municipal corporations, which generally consist of the people living within the corporation's territorial boundaries, exercising their collective will through their elected representatives, and assigned certain rights and obligations as a result of their collective status.
Although the same word is used, it would generally not be the case that a municipal corporation would be governed by the same laws as business corporations or trade guilds, which are also sometimes referred to as corporations. Barclay's and the City of London are both corporations, and therefore are treated as entities capable of making their own decisions, with rules allowing for their decision-making bodies to keep them "alive" in perpetuity. But there are also many obvious differences. The City of London does not have shareholders in the same sense that Barclays would, nor is it obligated to register its existence or undertake most of the other responsibilities facing a corporate entity.
Wikipedia has more detail broken down by jurisdiction.
A corporation is one type of judicial person
All corporations are creatures of statute; that is, they exist because there is a statute that creates them or allows them to be created. Most corporations in the UK are companies created under the Companies Act 2006; the City of London (formally the Mayor and Commonality and Citizens of the City of London) isn't one of those.
The statute that first created the City does not survive, and the city is regarded as being incorporated by prescription, meaning the law presumes it to have always existed (which is, obviously, not true - it's another legal fiction). The first recorded Royal Charter dates from 1067 when William the Conqueror granted the citizens of London a charter confirming the rights and privileges that they had enjoyed since the time of Edward the Confessor. Over the centuries, various charters and acts of Parliament have modified how the corporation operates.