No,databases simply store one aspect of the results of assessment.
Assessment, as you will know, is continuous, ongoing, mainly formative but with some summative elements - e.g. assessed tasks, tests etc.
The overall 'numerical' progress of pupils within a set or stream can be stored within a database and analysed.
The 'narrative' assessment is, within any multi-teacher, multi-subject school environment, known by individual subject teachers and shared formally and informally with someone with overall academic and pastoral responsibility for the child.
There is no significant difference between the 'setting' and 'streaming' versions of this, as far as I can see. In both cases, the 'statistical' and 'narrative' versions of the progress of each individual child in each individual subject need to be co-ordinated and interpreted by one individual - the tutor - or a small group of such individuals - e.g. tutor, head of house, head, SENCo etc.
Whether the 'feeds' into that individual / small group are from a subject teacher per stream or per set makes little difference - the fact remains that it comes from multiple sources and has to be co-ordinated. It is not like in primary, where a single class teacher teaches most subjects and aintains a clear overview of all subjects - in secondary it is all about collating many views, however classes are organised.