What is the implication of overlap between eigenstates of two operators in Quantum Mechanics? For instance, what does it mean that a certain position eigenstate is also an energy eigenstate? 
I understand that measurable (Observables) in Quantum mechanics are the operators. Their eigenvalues are the possible values that we can get in measurement. But, in general a system can be in states other than the corresponding eigenstates.
What isn't clear to me is what the overlap between eigenstates of two operators implies?
What does it mean physically?
Why do we care if such states exist?
 A: The mathematical formulation of QM is built upon operators.
Observables are physical quantities that can be measured.
The wavefunction represents the probability amplitude of finding the QM system in that state.


Any observable, i.e., any quantity which can be measured in a physical experiment, should be associated with a self-adjoint linear operator. The operators must yield real eigenvalues since they are values which may come up as the result of the experiment.
    In the wave mechanics formulation of QM, the wavefunction varies with space and time, or equivalently momentum and time (see position and momentum space for details), so observables are differential operators.
Let ψ be the wavefunction for a quantum system, and {\displaystyle {\hat {A}}} {\hat {A}} be any linear operator for some observable A (such as position, momentum, energy, angular momentum etc.), then
    {\displaystyle {\hat {A}}\psi =a\psi ,} {\hat  {A}}\psi =a\psi ,
    where:
    a is the eigenvalue of the operator, corresponding to the measured value of the observable, i.e. observable A has a measured value a
    ψ is the eigenfunction of {\displaystyle {\hat {A}}} {\hat {A}} if this relation holds.
If ψ is an eigenfunction of a given operator A, then a definite quantity (the eigenvalue a) will be observed if a measurement of the observable A is made on the state ψ. Conversely, if ψ is not an eigenfunction of A, then it has no eigenvalue for A, and the observable does not have a single definite value in that case. Instead, measurements of the observable A will yield each eigenvalue with a certain probability (related to the decomposition of ψ relative to the orthonormal eigenbasis of A).


https://en.wikipedia.org/wiki/Operator_(physics)
Now you are asking about the overlap between two wavefunctions.
According to the superposition principle, wavefunctions can be added together and multiplied by complex numbers to create a new wave function and a new Hilbert space.


The inner product between two wave functions is a measure of the overlap between the corresponding physical states and is used in the foundational probabilistic interpretation of quantum mechanics, the Born rule, relating transition probabilities to inner products.
    Physically, different wave functions are interpreted to overlap to some degree. A system in a state Ψ that does not overlap with a state Φ cannot be found to be in the state Φ upon measurement. But if Φ1, Φ2, ... overlap Ψ to some degree, there is a chance that measurement of a system described by Ψ will be found in states Φ1, Φ2, ... . Also, selection rules are observed to apply. These are usually formulated in the preservation of some quantum numbers. This means that certain processes allowable from some perspectives (e.g. energy and momentum conservation) do not occur because the initial and final total wave functions don't overlap.


https://en.wikipedia.org/wiki/Wave_function
Now for example in a covalent bond, the two atoms' have their own valence electrons and those have their own wavefunctions. When the covalent bond forms, the two wavefunctions overlap. The valance electrons of the atoms start existing around both atoms as per QM.
In QM, there are general rules about wavefunction overlap for bosons and fermions (symmetric and antisymmetric).
