First, you recognize that in many ways they are incompatible systems of typing, and don't necessarily try to merge or join them. However, you SHOULD probably have the same or similar type in both systems, because they are based on a lot of similar ideas. For example, it will be quite rare for someone to be an introvert in one system and an extrovert in the other.
Secondly, there are differing descriptions and ideas of the cognitive functions. Again, they are mainly the same, but different enough that what constitutes a function can be different depending on the source you use.
If you want to be pragmatic about it, you can stick with MBTI and not get bogged down with cognitive functions. Realize that there is a lot more you might get out of cognitive functions, but that it is quite the rabbit hole to go down. It depends on what your goal is with typing yourself and how interested in delving into this stuff you are.
Also, consider that online tests are not always an ideal way to find your type. Tests can only do so much, and are flawed. That they put you in some particular grouping does not mean that it is correct. I am often typed ENTP or INFP by tests, depending on a lot of factors but ultimately boiling down to the natural shortcomings of the tests.