In a strict sense, cats do not have periods primarily because they do not menstruate. A period is part of the menstrual cycle for female humans. This happens normally every 28 days or so. It can last between three and eight days. During this time, in response to reproductive hormones a woman’s womb gets ready for pregnancy. The endometrium thickens developing an extensive network of blood vessels. If the woman does not get pregnant this thick tissue sloughs off and causes bleeding which is menstruation.
What I have described does not happen with cats. The word “period” tells us that the menstrual cycle happens during a regular cycle. The time between menstruation is approximately the same.
Cats in contrast have repeated heat cycles over a year. The mating season for cats in the northern hemisphere is from March to September and in the southern hemisphere from October to March.
Queen’s (female cats that are not spayed) go into and out of heat several times throughout the breeding season. They do not always display oestrous behaviour at regular intervals. Cats are considered to be primarily induced ovulators, by which it is meant the act of mating causes the cat to ovulate. The cat goes through four distinct stages in the oestrous cycle. I won’t describe them in detail here but suffice to say that at no time during this cycle does the cat menstruate like a human being (read about cats in heat).
As menstruation is an essential part of periods, it cannot be said that cats have periods.
Although there are certain similarities between cats and humans in terms of preparing for pregnancy, the differences are too fundamental, I would argue, to allow me to say that cats have periods.