Two diagrams may be different, but they may still have the same isomorphic limits (or colimits).
Ind-objects are, so to speak, formal colimits of diagrams, even if the actual limit may not exist. How can they then be isomorphic? How is an isomorphism defined for general ind-objects?
If $"\varinjlim" \alpha$ and $"\varinjlim"\beta$ are two Ind-objects over a category $C$ then they are in particular contravariant functors from $C$ to $Set$. Actually we define the category $Ind(C)$ as a full subcategory of $C^{\wedge}=Fct(C^{op},Set)$ with ind-objects as objects and natural transformations as morphisms.
Hence $"\varinjlim" \alpha$ and $"\varinjlim"\beta$ are isomorphic if there are natural transformations $F : "\varinjlim" \alpha \to "\varinjlim"\beta$ and $G : "\varinjlim" \beta \to "\varinjlim"\alpha$ which are inverses of each other.