Looking up dictionary definitions of algebra geometry is pretty unsatisfying as they are usually along the lines of
"the branch of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations."
"the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogues"
which don't answer the question much further than 'geometry is what geometers do' and an even more cringeworthy 'algebra is black scribbles on a page for people doing algebra'. The only real difference I can find is the reference to quantities (but what is a quantity?). There are also often remarks about geometry referring to 'real world' objects such as shapes and solids and relations between them. I'm hesitant to accept these because they fail to recognise the distinction between observations and mathematics.
So to ask the question more precisely, how do you take a set of axioms and know they describe a 'geometry' or an 'algebra' or any other subject for that matter? It seems unlikely that mathematicians would label things differently without having a clear distinction between them. To the best of my understanding, algebra defines operations on 'things' and geometry is to do with how 'things' stay the same. Now that I say it it doesn't make much sense. Any help?