Being aristocrat isn't the same as being of noble birth. Aristocracy implies a ruling group. Nobility only rules today in a number of (more or less absolute) monarchies.
Only as a system of government - 'Britain is an aristocracy' would mean that Britain was ruled by its noblemen, but 'Britain has an aristocracy' simply means that noble families exist in Britain.
Or for "democracy", by the way.
Indeed - after chewing this over, I'm convinced by Dachs' argument that 'absolute monarchy' as a term doesn't mean very much, but disagree that this makes it useless. Although we can't characterise a society by it alone, because it lumps together 1640s England, 1780s France and 1890s Russia, it is useful to get a feel for a society in conjunction with other descriptions, and as an opposite to 'constitutional monarchy', which is a phrase which lumps together 1790s France, 1910s Russia and 1990s Britain, and so is equally poor as an overall descriptor of a country. It's as useful as terms like 'large income inequality', 'highly stratified society' and 'militaristic society' - although neither of these are great in isolation, if we talk about 'an absolute monarchy with a highly stratified society, large inequality of income and a dominant military', we begin to get a feel for the society being described.