That’s right - Adam & Company is another brand under the RBS umbrella, based in Scotland, which is run independently too.
As @Recchan says of Child & Co, the legal connection is due to their proximity, especially to prominent Inns of Court, so they seem to be well-used by barristers. That creates a network effect which is probably part of the reason they also have a large customer base in the larger legal field.
A Child & Co private banking account would be more akin to Coutts than RBS, the personal version we have is just for fun really.