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In Aluffi's book Algebra, just from possibility of writing an abelian group G isomorphic to $langle grangle oplus G/langle grangle$ it concludes that by induction $G cong mathbb{Z}/d_1mathbb{Z} oplus dots mathbb{Z}/d_nmathbb{Z}$ with specific rules on $d_i$'s!

1- So why not $G cong mathbb{Z}/p_1mathbb{Z} oplus mathbb{Z}/p_1mathbb{Z} dots mathbb{Z}/p_1mathbb{Z} oplus mathbb{Z}/p_2mathbb{Z} dots mathbb{Z}/p_2mathbb{Z} oplus dots dots dots mathbb{Z}/p_nmathbb{Z}$?

2- Why $d_i|d_{i+1}$? (a must/theorem or a choice/standard-rule?)

3- And if writing them as powers of primes so why following the below steps?

I'm assuming you're talking about finite abelian groups, otherwise one would need a free group $mathbb{Z}^r$ in the decomposition.

1) We may not have necessarily have prime orders. Note $<g> cong mathbb{Z}/dmathbb{Z}$ if $g$ has order $d$ in $G$. As an example, $mathbb{Z}/4mathbb{Z} notcong mathbb{Z}/2mathbb{Z} oplus mathbb{Z}/2mathbb{Z}$.

2) The order just gives a standardized way of representing the group. One can show that the representation is unique up to order of the terms in the direct sum. If one writes the invariant factors in this way, there is an algorithm to compute the elementary divisors. Similarly, given the elementary divisors, the algorithm gives the invariant factors in such a way that the divisibility condition holds

3. Elementary divisors and invariant factors are different things. The elementary divisors are a decomposition of the group into $primary$ ideals, that is power of prime ideals. One can determine these primary ideals given the invariant factors using the process described above.