It's become common to talk about Pets versus Cattle as the "new way" of thinking about servers.
Of course, "the new way" isn't really new - many IT shops in the mid 1990s had fully automated, reproducible, and disposable infrastructure. It's just the term that has recently become trendy, and I don't think the analogy is necessarily right.
In the original analogy, the claim was that a Pet is precious, so you care and feed for it specially. If it's sick, you nurse it back to health. Whereas if one of your herd of Cattle gets sick, you take it out back and shoot it. This is based purely on emotional attachment, and makes little business sense. The truth is more that most Pets have little financial value, whereas Cattle are intrinsically valuable. Whether sick Cattle are bursed back to health should be a pure business decision based on the value of a healthy animal compared to the cost of treating it.
Currently, I think a more appropriate analogy would be Cats versus Petals.
Let me explain.
A Cat system has a mind of its own. In fact, it isn't at all clear whether you own the system or the system owns you. Cat systems tend to be solitary and not integrate or interoperate well with others. If you have many Cat systems, they will tend to wish to go their own ways.
In contrast, Petals will be small, simple systems. You will have many, and they will be the same. While a Petal may have some value of its own, their true beauty is only visible when they are put together into larger units - flowers, for example. Different flowers are made up of different types of petals.
One point here is that if you're thinking about Pets and Cattle, you're still thinking of individual animals. With Petals, the role of holistic thinking and orchestration in producing a larger object (the flower, or even the garden) becomes clear.
In terms of terminology, your business is a garden; the services you provide are flowers; they are constructed from containers as the petals via an orchestration service that provides the stems and branches. Your job is to ensure good soil, water and light, prune, remove pests and weeds - not to create each individual Petal by hand.
If you're still herding Cats, it's time to stop and tend gardens instead.