I love my cat with all my heart, but she's, to put it lightly, a bit of an asshole. Earlier today I was walking past my toilet, and she scratched my toe because while stretched out across the floor I apparently invaded her personal space. It bled enough for me to decide I should probably to splash the cut with hydrogen peroxide and put a bandage over it. It got me thinking about how apparently minor cuts could get rapidly infected and be extremely fatal before modern medicine, germ theory, sanitation, proper wound cleaning, etc.
Exactly how bad was it, and if it was as bad as people/pop culture would make it out to be, how did we interact with our pet cats if even minor cuts were so incredibly dangerous? If they were as dangerous as people make it out to be, it's a little hard to believe we would tolerate the adorable bastards while playing constantly cutting us.
Hey, so I wrote an answer to a similar question here that I hope you might find useful. The source I go to is Bald's Leechbook, a 9th Century English medical textbook probably written during the 'Alfredian Renaissance' and named after its patron. The Leechbook contains a combination of Classical and British/English medicine, and a wide array of herbology. While your average 9th century physician might not have been intimate with germ theory and the modern scientific method, the Leechbook is in it's own way scientific, based on observable outcomes and clinically valid botany in many cases. While not as effective as modern medicine, recent testing has shown genuine efficacy for some of its treatments, and much of its advice is still valid medicine.
Despite the lack of a coherent germ theory, the Leechbook did understand the importance of cleanliness in dressing and treating a cut or wound, as well as in treating any subsequent infections. The efficacy of this is of course dependent on the circumstances of the wound and the rapidity with which you might seek medical aid. The suggested courses of action are detailed in the answer I linked above, there is essentially a whole section dedicated to:
Treatments and wound salves and drinks for all wounds and cleansings in every manner
The majority of these treatments require herbal remedies, but a central component of all of them is either that everything is boiled together, or added to hot wine; while these practitioners may not have understood the concept of sterilisation, they clearly understood that wounds washed in boiled water or alcohol were much less likely to be infected.
Anglo-Saxon medicine is big on topical treatments, salves and washes. Many of the salves and washes listed in the Leechbook contain herbs still used today (albeit in clinically refined form) as anti-inflammatories, analgesics or anti-fever medications, and it even contains a primitive coagulant to try and stop hemorrhaging wounds. A small wound like your cat scratch could be treated with warm, clean honey (again known for antiseptic properties) or:
If you want to quickly treat a small wound, triturate watercress or boil in butter, make into a salve, smear with.
It's likely that the washing with boiled water is the most important part of this treatment. In the event that the wound gets infected, the Leechbook suggests washing it with hot wine, cutting away dead flesh and binding with honey. There are treatments even as far as amputation of infected and gangrenous limbs.
This doesn't all mean that a scratch and infection wouldn't be dangerous. Cat bites and scratches can still be dangerous today because of their difficulty to effectively clean, and sepsis is a killer even with modern medicine. However, you would have still understood the importance of cleaning the wound and contemporary medicine would have had treatments that they understood to be sufficiently effective to use regularly.
In the context of peoples in North America I'll quote the relevant section from an earlier answer of mine:
Snakebites and spider bites were treated by the cupping and suction method, where the bite site is lanced and the poison sucked out with a tube. This method was also used for other topical infections by North Americans....Some amount of surgery was done such as simple plastic surgery, thoracentesis, and amputations; with wounds sewn together using human hair sutures. Anishinabe people would perform surgery on torn ears by first trimming the tissue then suturing the sides together evenly. Great Lakes peoples such as the Othaakiiwaki (Sauk) and Meskwaki practiced thoracentesis, which is a surgical procedure where the chest is punctured allowing excess fluids to drain. This was done to treat empyema. Some North American healers used amputations with flint knives, cauterizing the wound with a hot stone. North Americans, as well as Mesoamericans and South Americans made skin grafts (sadly I don’t know more about this).
Wounds were treated with poultices, such as those made of datura by Virginia Algonquins. The Makah people used yarrow as an antibiotic for treating wounds, and the Yurok used rhizomes of a fern (polypodiaceae) for the same purpose. A:shiwi (Zuni) people used sage to treat foot infections, and peoples in the eastern woodlands used cranberries and blueberries to treat infections. Cleaning wounds by removing dead tissue was done by the Aztec peoples and other North American peoples. Wounds would be dressed with a clean cotton bandage, which was changed daily. Sometimes the wound was packed with eagle down, a puffball (a fungi filled with spores that resemble dust, used by plains peoples), or spider webs (used by the Kwakiutl). Alum was used by the Chickasaw and Meskwaki on wounds, and the Iswa made a powder of dried corn smut to stanch bleeding.
Wounds were flushed with syringes or with a drain by some peoples. The Inday (Mescalero Apache) treated deep wounds with drains made of slippery elm or twisted cotton. One was placed at the bottom of the wound and sutured in place, these were removed after about nine days. The Inday, Dakota, Ho-Chunk, and the Scaru:re? (Tuscarora) all used drains or hemostats. Various peoples made drawing salves, which were applied to infected wounds to help them drain; such as the Neshnabe and Shoshoneans. The Neshnabe and others used pine tree resin, which was first warmed then applied to the wound and bound to it with a strip of leather. The heat served as a vasodilator, helping the draining process. Other peoples used white pine for this purpose.
In terms of knowledge of medicine, it vastly depended on the time period and the continent. Human knowledge of medicine wasn't widespread amongst a worldwide community like it is today. Prior to the 20th century, medical knowledge was very much isolated to the continent you were living on.
In 800 BC the Indian physician Sushruta practiced and wrote about thoroughly cleaning and sterilized deep wounds before applying stitches. His texts also write about successful surgical procedures to remove blood clots.
Bhattacharya, Surajit. “Wound Healing through the Ages.” Indian Journal of Plastic Surgery : Official Publication of the Association of Plastic Surgeons of India, Medknow Publications & Media Pvt Ltd, May 2012, www.ncbi.nlm.nih.gov/pmc/articles/PMC3495363/.
Pre Columbian Americans in both north and south America created and used syringes and hypodermic needles made from bird bones to inject drugs and to clean out deep wounds. Similar methods have also been found in Africa and in Australian Aboriginals.
Kiger, Patrick J. “10 Native American Inventions Commonly Used Today.” History.com, A&E Television Networks, LLC, 14 Nov. 2019, www.google.com/amp/s/www.history.com/.amp/news/native-american-inventions.
In terms of Europe, medicine was pretty on par with the rest of the ancient world in Ancient Greece and up to the Roman Empire but took a regressing turn from the Dark Ages to the end of the Victorian Era, (when bathing and personal hygiene began to improve and thus decreased the amount and severity of infections from minor injuries) up until that point, hygiene was not seen as connected to health at all. Hand washing was not widely practice among medical professionals until the 1850s and proper anal hygiene was not widely practiced until the 1890s when the toiler paper roll became available in 1891. Because of this, minor injuries often became infected and treatment focused on treating the infected tissue and patient versus practicing any preventative measures to stop the secondary infection.
Heichelbech, Rose. “What Hygiene in the Victorian Era Was Really Like.” Dusty Old Thing, Great Life Publishing and Greater Good, 23 Oct. 2019, dustyoldthing.com/victorian-era-hygiene/.
“Historical Perspective on Hand Hygiene in Health Care.” WHO Guidelines on Hand Hygiene in Health Care: First Global Patient Safety Challenge Clean Care Is Safer Care., U.S. National Library of Medicine, 1 Jan. 1970, www.ncbi.nlm.nih.gov/books/NBK144018/.
So all in all to answer your question, if you lived prior to the Dark Ages, or with some indigenous people that's used to getting cuts here and there, you would probably be ok. If you lived in Europe from the Dark Ages to the end of the Victorian Era, you would have a much tougher time.