I don't think there is anything to worry about here. BSocSc is just the designation that UoM (University of Manchester) uses for most of its social science degrees. It doesn't mean anything different to BA or BSc. And lots of Universities actually use something quite specific, for example Cardiff uses BScEcon - even for Sociology, Politics etc.
Now, it gets a bit confusing because some social science degrees at Manchester are 'BA' - when a social science subject is combined with something from Arts and Humanities, typically a language
Note that 'social science' relates to subjects in the School of Social Sciences at this University (so at UoM, Philosophy and Economics are classed as Social Science where they might not be elsewhere)
And it gets even more confusing with the 'BASS' programme. This is actually a specific degree programme and pathway that is intended to allow students to study a range of social science subjects in their first year, and then specialise in years 2 and 3. And so on one level, it is an acronym. But IIRC, the degree is also labelled 'BASS' and reflects a combination of two social science subjects (e.g. Sociology and Politics)
Again, though, I wouldn't worry too much about BA vs BASS vs BSocSc
Manchester is incredibly well respected in this area. The Sociology department is by most accounts the best in the country and one of the best in the world. The other social sciences are similarly well regarded. I don't imagine that many employers would concern themselves with this, rather, I think they would be looking for a 'good' or 'prestigious' Uni - which Manchester definitely is. And whilst it isn't Oxford or Cambridge, most would know that Manchester is one of the best.
I hope this helps. But the bottom line is that I don't think there is anything to worry about and you/DC will have a well respected degree at the end of it (and probably a good time living in Manchester along the way)