Personality is the sum collection of all life experiences that have contributed to us being a one of a kind individual. Personality is broad and includes our persona, our character, our emotional defenses, our likes, wants and fears, our cognition, and our memories. Thus, by definition, personality is not completely innate; personality develops.

Cognition is a sub-set of personality. It's the part of us that deals with our neural hardware; our method of metabolizing information, regardless of what that information is. Cognition is content-less, in that it possesses no specific opinions, beliefs or values in itself. However, it's a mode of thinking which might commonly produce opinions, beliefs and habits, but it is not defined by them.

Not everything can be explained as being due to cognition or cognitive type, and one shouldn't expect their type to be an answer to every idiosyncrasy of character a person possesses. Life experiences, choices and interests and contribute to whole personalities being similar or different to each other.

If a system classifies people based on personality profiles and personality similarities, then that system will group together anyone who relates to a given psychological portrait, regardless of what the underlying reason is - cognitive or otherwise. The same personality traits can be generally arrived at from different cognitive origins. For example, two individuals may be unsocial, but for one it's because they are anxious and have low self esteem, while another individual avoids people because they find them uninteresting and prefer mechanical topics. These are very different underlying causes and point to two very different people underneath, despite the same personality trait of "unsocial" being arrived at from different angles. 

Typing by personality will allow people to relate to others at a behavioral level, but will not connect people together who "think" the same way.

If instead a system classifies people based on cognitive similarities, then that system will group together anyone who metabolizes information the same way, regardless of their overall personality similarities or dissimilarities. This grouping will be less homogeneous at the behavioral level, and may include people who don't always "relate" to each other personally, but who nonetheless "think" the same way.

They may have different ideas and opinions about things, but what's important is that the manner in which they form those opinions and ideas carries within it the signature of the same cognitive processing style.

Since Cognitive Typology (CT) types based on the vultology that emerges from real-time cognitive processing, a typing by CT may not match you necessarily to people you most "relate" to at the individual level, since relatability happens from all elements of personality. CT does provide behavioral approximations, based on what personality traits are more statistically common to a given psychology, but these cannot be used as definitive criteria for or against being of a given cognitive type.

While it may be valuable to rely on personality models for a more specific kind of self-understanding, that self-understanding will not be about anything fundamental, but about thing that are contextual to you. A person can be grouped together with others who share a particular temperamental "fit", but that fit can be temporary, caused by life choices, accidents of upbringing, and can change over the course of life. 

Oppositely, a cognitive system aims to describe something about you that is innate, unchanging and always true. Even if you evolve in your personality and change your opinions and habits about a lot of things, you would still do so from within the cognitive architecture you possess. We believe this is a much more stable grounding point for personal identity than a personality theory.

Yes. Personality and cognitive systems appeal to different people for different reasons. If you seek to be grouped with others who developed the same set of beliefs, ideas or habits as you did, it's best to look for a personality model. If you want to be grouped with others who share your innate modus operandi, it's best to look for a cognitive model. 

You can also do both, so long as you keep them separate and understand the two are not talking about the same thing. For example you might want to adapt an INFJ personality label in the standard MBTI, while also having an SeFi cognition. However, if your view of the MBTI is as a cognitive model (i.e. JCF) rather than a personality model, then a decision may need to be made as to which system is more accurately describing your cognition, and the two cannot coexist in a coherent form. 

Below is an example of how one might address each aspect of our personality, with popular models that address the same areas:

To get the closest approximation possible of personality as a whole, multiple different systems are needed to cover different areas of focus. An example of such a combination may be: SeFi / INFP / type 9 / ADHD

But no system should be used as a cover-all for every aspect of personhood. Personality is too vast and complex to be determined by any single factor. Humans are much more complicated than that, and a more nuanced understanding is needed, to get a full handle on who we are.