Grey Wizards (among others) are able to create familiars out 'mud, clay, dung, blood, and the vital organs of humans and animals, but these only scratch the surface'. It would take about a month and a great deal of money to gather all the components and between one and three to piece them together and instil them with your magic, and if you make a mistake you'll have to go back and start instilling it all over again. At the end you'll have a homonculus or golem made of wood, stone, flesh, clay, or whatever else, based on either humanoid or animal design. It would be obviously artificial to look upon and usually varying from the base form of the creature it resembles in some way. It would also have a degree of sapience, the ability to communicate (verbally or otherwise), and some measure of free will, though you can magically impose your will to force it to obey. You will have little control over what sort of relationship your familiar will have with you when first 'born', though you can hope to change it over time. You're also not yet magically skilled enough to attempt this, though learning enchantment would be a shortcut to allow it.
Grey Wizards, like Bright and Light Wizards, are one of those fortunate orders able to both create and bind familiars. Binding a familiar means either searching for months or spending an outrageous sum of money for an animal with that certain je ne sais quoi, a combination of health, appearance, attitude, intelligence, as well as an intangible form of potential within it. Then you need to go full crazy cat lady and spend no more than a couple of hours a day separated from your new baby for a period of a month or two, though you can be doing other things as long as you're doing it in the immediate presence of your would-be familiar. At the end of it, if you've formed enough of an emotional and mental bond with the creature, they'll have a mental awakening and become your familiar. It's personality and level of obedience will vary quite a lot based on random chance and the animal itself - a dog is more likely to be eager to please than a cat is, for example - but like with created familiars, you can mentally impose your will over the creature.
There is no limit to the amount of familiars you can have except for money and time.