Why a Razor?

Seppuku

Also known as hara-kiri ('hari kiri'), this traditional practice is a form of Japanese ritual suicide by disembowelment. It was originally reserved for samurai. Part of the samurai bushido honor code, seppuku was used either voluntarily by samurai to die with honor rather than fall into the hands of their enemies (and likely suffer torture) or as a form of capital punishment for samurai who had committed serious offenses, or performed because they had brought shame to themselves. The ceremonial disembowelment, which is usually part of a more elaborate ritual and performed in front of spectators, consists of plunging a short blade, traditionally a tantō, into the abdomen and drawing the blade from left to right, slicing the abdomen open.

A tantō. Could have been used for shaving. Maybe.

The first recorded act of seppuku was performed by Minamoto no Yorimasa during the Battle of Uji in the year 1180. Seppuku eventually became a key part of bushido, the code of the samurai warriors; it was used by warriors to avoid falling into enemy hands, and to attenuate shame and avoid possible torture. Samurai could also be ordered by their daimyo (feudal lords) to carry out seppuku. Later, disgraced warriors were sometimes allowed to carry out seppuku rather than be executed in the normal manner. The most common form of seppuku for men was composed of the cutting of the abdomen, and when the samurai was finished, he stretched out his neck for an assistant to sever his spinal cord by cutting halfway into the neck. Since the main point of the act was to restore or protect one's honor as a warrior, the condemned should not be decapitated completely but only halfway. Those who did not belong to the samurai caste were never ordered or expected to carry out seppuku. Samurai generally could carry out the act only with permission.

https://en.wikipedia.org/wiki/Seppuku

During the Victorian era Europeans were fascinated with stories of travel and exotic rituals. It is highly probable that Strindberg (as well as his audience) was familiar with this Oriental practice.