The Blackfriars of this campaign refers to the area around Blackfriars Road, on the south side of Blackfriars Bridge in London SE1. The road gets its name from Blackfriars Bridge, the first version of which was opened in 1769. The bridge got its informal name first from the Blackfriars Monastery, a Dominican priory that stood nearby, in the City of London on the northern side of the bridge. Blackfriars Road was originally built in the 1760s as the southern approach to the new bridge, and was originally called Surrey Road. Now the area in the north-western corner of the Borough of Southwark, and even parts of the north-eastern Lambeth around Southwark underground station is often informally known as Blackfriars. Officially the area belongs to the Borough and Bankside Area of Southwark.
The Southwark Council Planning Character Appraisal for the Blackfriars Road Character Area (.pdf) is online on the council website here. Incidentally it does not list the proposed Linden Homes site at 169-173 Blackfriars Road as nominated for development. It does however recommend a listing for the Sons of Temperance building at 176-177 Blackfriars Road, adjacent to the Linden Homes proposal site.
The north part of Blackfriars Road, between The Cut and the river Thames is also designated by Southwark Council as suitable for tall buildings. The Linden homes development is outside this area, even if one of the proposals is for a 27 storey building. The Council Tall building study (.pdf) is available here.