This implement is stereotypically used in domestic spanking contexts such as mother/son or mother/daughter, in cases where a harder punishment than a hand-spanking is intended. In vintage photography the hairbrush is mainly seen in series where ladies in lingerie are pulling other woman over their knees for a hairbrush spanking.
I found some very special series with anonymous models getting spanked with a hairbrush. Of course other types of brushes were used as well, like the bathbrush.
Spanking with a paddle is known as paddling. This implement was mostly known for punishment at school (Us mostly) A known paddle was the Knoppler paddle, according to Mistress Penny from Strictly Leather, a British BDSM and spanking toy maker, it is a short leather implement “that was used to administer punishment in schools and prisons in Germany and nearby countries. It is usually constructed from leather but wooden versions have also been seen. The use of the knoppler paddle has now spread worldwide and was adopted extensively in North America possibly because of immigration from Europe during the early 1930s.