Neither are really part of Birmingham mate. Solihull is a separate borough and whilst Sutton falls under Birmingham City Council it very much has its own identity - people who are from there tend to say they are from Sutton, not Birmingham. Don’t really like either tbh, certainly not Solihull which I feel lacks personality. Both places are where self made people tend to gravitate though, hence my comment.
The nicest areas in Birmingham proper are Harborne and Edgbaston I’d say, the former of which is most akin to Didsbury. The latter has some ridiculously big houses. As does Moseley, although demographically it’s more mixed in that district I’d say.