Mini breeds are sometimes used in agriculture due to being efficient producers of meat and milk on small lots of land, or in places where available fodder isn't very rich - see the Australian Lowline or the Dexter which are production animals and not pets. These are usually about a metre (3') tall at the shoulder and weigh at least a couple hundred kilos, so they're big animals despite being small cows. Cows smaller than this (like a micro mini) usually have some form of dwarfism like achondroplasia which humans also get, and which can cause health problems.
But even in a perfect world where cows could be infinitely tiny without health problems, I would still strongly be against keeping a cow as a house pet for multiple reasons:
Cows have different social needs from cats or dogs because they're herd herbivores. To feel safe and complete, they need to be with other herd herbivores - cows preferably but they can get along with sheep, horses, goats, and others. Cows can be great friends with humans but humans can't be their world.
The ruminant feeding cycle isn't suited for being a house pet. Cows typically graze continuously at dawn and dusk and nibble throughout the day, finding cozy spots in the sun to ruminate between bouts of feeding. This is awkward to set up in a household environment because it requires a lot of space and sun.
Cows (and other herbivores) don't understand bowel control like cats and dogs do. They will urinate and defecate wherever and don't understand why you would try to stop them.
Because of their complex social needs, they can develop serious behavioural issues if reared away from other calves, but if you rear them with other calves and then isolate them in a house, you're also heading towards trouble. There's no right time to take cows away from herd living.
Pets like dogs and cats enjoy being in a house and having human friends, and they benefit a lot from this. For a cow, you're just compromising their needs (open space, cowie friends, lots of grass and other fodder available) to suit your own pleasure. This is a terrible way to think about your pets, and having a housepet should be just as fun for them as it is for you.
For all of these reasons I'm strongly opposed to keeping any cow alone and especially alone inside a house. They're not dogs and they're just not made for it! Love your cows how they are and not because of how much they could be like something else.