Q.1. Bruns_Herzog define multiplicity (in the case of graded rings and modules) as
My question is that:
why multiplicity for $d=0$ it is defined as $\ell(M)$? Is there a kind of compatibility with the case $d\gt 0$?
Edit.
Q.2. When the ring and module are not necessarily graded, and $I$ an ideal of denition of $M$, they define multiplicity of M with respect to I as $e(I,M)=e(gr_I(M))$, see 4.6. Now the same question arise:
why multiplicity in this case is defined as multiplicity of associated graded module? Is there a kind of compatibility with the case of graded rings and modules?
Let me give you another point of view on multiplicity.
Remember that a function L on the category of right R-modules and taking values in $\mathbb R_{\geq0}\cup\{\infty\}=\mathbb R^*$ is a length function if it is additive on short exact sequences and it is continuous on injective direct limits, that is, $$L(A)=L(B)+L(C)$$ if $0\to B\to A\to C\to 0$ is short exact, and $$L(M)=\sup_{i}L(M_i)$$ if each $M_i$ is a submodule of $M$, the system $\{M_i:i\in I\}$ is directed and $\sum_iM_i=M$.
Our main example of length function is the composition length $\ell$ of modules.
Let me recall you also some theory of modules over rings of polynomials. Indeed, a right $R[X]$-module $M_{R[X]}$ is just a right $R$-module $M_R$ with a distinguished endomorphism $\phi:M\to M$ that represents "right multiplication by $X$". Similarly, a right $R[X_1,\dots,X_k]$-module is nothing but a right $R$-module with a $k$-tuple of pairwise commuting endomorphisms acting on it.
We can now return to your setting. Indeed, $R$ is a commutative Noetherian ring, $M$ is module over $R$ and $I=(x_1,\dots,x_k)$ is an ideal of definition of $M$. Denote by $\phi_i:M\to M$ the homomorphism of $R$-modules that consists in multiplying an element of $M$ by $x_i$. This gives $M$ a natural $R[X_1,\dots,X_n]$-module structure (notice that for this construction you do not need $R$ to be commutative but you need $I$ to be generated by a set of central elements of $R$, you need Noetherianity to say that $I$ is finitely generated).
Now we can define a multiplicity. We proceed inductively, in fact it is convenient to define multiplicity attached to a general length function, not just to $\ell$. So, let us choose a length function $L:Mod(R)\to \mathbb R^*$ and let $F$ be a finitely generated $R[X]$-module ($R$ Noetherian commutative), and let $\phi:F\to F$ be the endomorphism representing the action of $X$ on $F$. Then, $$mult_L(F_{R[X]})=\begin{cases} L(F/\phi(F))-L(Ker(\phi))&\text{if $L(F/\phi(F))<\infty$;}\\ \infty&\text{otherwise.} \end{cases}$$ One can now define $mult_L(M_{R[X]})$ to be the supremum of the multiplicities on the finitely generated submodules of $M$. In fact we can show that $mult_L$ is a length function of the category $Mod(R[X])$. Proceeding inductively (that is, you can define now a multiplicity of $Mod(R[X_1,X_2])\cong Mod((R[X])[Y])$) you can define a multiplicity of $R[X_1,\dots,X_k]$-modules and show that it is a length function, of course there are many things to check.
The thing you should check is that in this way you obtain the same value given by the definition in your question, this is not difficult but not completely trivial.
If you arrived until here, the answer to your first question should be clear: you have a sequence of multiplicities, where the $0$-th multiplicity is just the length function $\ell: Mod (R)\to \mathbb R^*$ and, for all $k\geq 0$, the $(k+1)$-th multiplicity is defined as follows: we let $L_k:Mod(R[X_1,\dots,X_k])\to \mathbb R^*$ be the $k$-th multiplicity, then $$L_{k+1}=mult_{L_k}:Mod(R[X_1,\dots,X_k][X_{k+1}])\to \mathbb R^*.$$ I think that this is the "compatibility" you were looking for.
Let me finish saying that the material that I have tried to briefly explain here is not original but it comes from Northcott's book "Lessons on Rings, Modules and Multiplicities" and from Peter Vámos PhD thesis "Length functions on modules".