The Kalashtar features come from a spiritual link; Wild Shape does not sever this link
The Wild Shape feature states:
[...] You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so. [...]
So we only need to know whether a given new form is physically capable of using Kalashtar features so let's look at where those are from:
[...] The kalashtar are a compound people, created from the union of humanity and renegade spirits from the plane of dreams—spirits called quori [...]
[...] Every kalashtar has a connection to a spirit of light, a bond shared by other members of their bloodline. Kalashtar appear human, but their spiritual connection affects them in a variety of ways. [...]
Nothing in the description of the Kalashtar gives conditions for this spiritual link to end, thus we can conclude that Wild Shape does not end this link and so Wild Shape does not stop the Kalashtar from benefitting from all these features.
Wild Shape also states:
[...] you retain your alignment, personality, and Intelligence, Wisdom, and Charisma scores. [...]
All of these things (as well as skill proficiencies and perhaps languages) are various aspects of the PC's knowledge and mentality and they are maintained during Wild Shape. Similarly, it would make sense to me if a spiritual link to your own mind were maintained during Wild Shape.