Can tweaking a cipher have the same or similar effect as changing the key? If we encrypt a message $m$ with (say) $AES$ and key $k$ we get ciphertext $c$.
And if we encrypt the same message $m$ but with $k^*$ we get ciphertext $c^*$:

1) $AES[E_k(m)] = c$ and $AES[E_{k^*}(m)] = c^*$ 

But what if we were to tweak $AES$ insted of changing the key? For example, a simple tweak might be to change the ShiftRow from right shifts of $0,1,2,3$ to (say) right shifts of $1,3,0,2$. And perhaps unique shifts for each round of the $10$ rounds.

Then, if we encrypt a message $m$ with (say) $AES_{tweak1}$ and key $k$ we get ciphertext $C$. And if we encrypt the same message $m$ with $AES_{tweak2}$ and with the same key $k$, we get ciphertext $C^*$:

2)  $AES_{tweak1}[E_k(m)] = C$ and $AES_{tweak2}[E_{k}(m)] = C^*$ 

Since these ciphertexts in (2) are distinct is this the same or similar to changing the key, as in case (1)? 

Let's assume sender and receiver can secretly exchange the tweaks.
Well it'll change the output for sure.  But it's not a great idea.  I don't know what exact arrangements you have in mind, so I'll think of my own along those lines and throw some numbers at you:

Possible number of 128 bit keys = 2^128 ~ 10^38

Possible number of ShiftRows tweaks = P(4!,10) ~ 10^13 -> 42 bits

So  better than I expected (42 bits) but still wholefully inadequate today.  However I'm not sure how you'd actually exchange the tweaks.  Little tables of shifts that you enter into a program that enters them into AES?  Clearly you can't use a password + standard key derivation function.

I'm always messing around primitives to create weird stuff.  But one thing that I've learned from this forum is not to mess with their internals.  They are carefully designed (some might say evolved over time) to be secure and efficient.  By tweaking them you can unwittingly fall foul of the law of unintended consequences.  What if a pattern is created?  Or some form of weak shifts?  Recall totally unforeseen RC4 weak states.  I'm not aware of any crypt-analysis of AES with your ShiftRow arrangement.  How do you know it's secure, just because you can't break it?  

So in summary it's not like changing the key to standard AES as that's how it's meant to be used.  You might enter a particular ShiftRow permutation and just break the thing security wise.