Each character has 4 companion slots to start. For premium currency (blue crowns), four more slots can be opened. It is possible that characters in excess of four can be stored at a personal homestead location called a concession, but I have yet to verify the mechanics.
Character progression puts limits on recruiting companions. Every character has an authority stat which is base leadership value without any modifiers. Humanoid companions have a defiance stat. Authority minus Defiance may not be less than zero.
Characters with the fauna skill also have a control stat. Tamed animals have a ferocity stat. Control minus Ferocity may not be less than zero.
Here's a basic companion, from the game guide.
(1) is the generic name, suitable for including in message board discussion
(2) is the portrait, which is the icon for interacting with the companion in game
(3) is the value of default skills, more on this in a moment
(4) is the list of skills that a companion has. Each type of companion has a tiny subset of the skills and stats available to a player character.
This is how my companion looks in game. Note the differences. The list of skills (1) is the same. Inspect the base values (3) some of them are lower than the default and some are higher. It may be worth trashing a companion with low values across the board in favor of a shot at something better. Therian Saga forums suggest that the random values are +/-3
The current value (2) of skills can be increased by a maximum of 15 points. To do this, the companion must be at your location while you perform a task using that skill. Setting a companion to gather materials counts as not at your location. As explained in other posts, the weighted average of Skill, Tool, Companion, and Location Bonus is used whenever a skill is checked. Thus, a companion with a high skill may make up for deficiencies in other areas.
Companions higher than tier one are gated by "Wings" which are expert level craft actions in Hawkoria. Some companions will only appear for recruitment at player constructed inns on player owned concessions.
When recruiting, each companion has a "presence" value from 0 to 100%. This is supposed to represent the amount of time that companion spends at the recruitment location available to be influenced by the player. As a practical matter, it means the player spends more actions recruiting a companion with low presence value. Some identical companions may have different presence in different recruitment locations.