Let $\mathcal A$ be the set of all $n\in \mathbb N$ with a prime factor $p>\sqrt{n}$. First, the number of elements $a\in \mathcal A$ with $a\leq x$ is $\gg x$. Indeed, this large prime factor is unique for any $a$, for a given $p\leq x$ there are $\left[\frac{\min(x,p^2)}{p}\right]$ integers $a\leq x$ with $p>\sqrt{a}$ and $p\mid a$. Therefore,
$$
|\mathcal A\cap [1,x]|=\sum_{p\leq x}\left[\frac{\min(x,p^2)}{p}\right]=\sum_{\sqrt{x}<p\leq x}\frac{x}{p}+\sum_{p\leq \sqrt{x}}p+O(\pi(x))=
$$
$$
=x(\ln 2+o(1))\gg x
$$
On the other hand, for any $\sqrt{x}<p\leq x$ all the numbers between $\sqrt{x}$ and $x$, which are divisible by $p$, lie in $\mathcal A$. Therefore,
$$
r_p(x)=\sum_{p\mid n, n\in \mathcal A} 1-\frac{x(\ln 2+o(1))}{p}\geq
$$
$$
\geq \left[\frac{x}{p}\right]-\left[\frac{\sqrt{x}}{p}\right]-\frac{x(\ln 2+o(1))}{p}=
$$
$$
=\frac{x(1-\ln 2)+o(x)}{p}.
$$
Summing over all $\sqrt{x}\leq y<p\leq x$, we get
$$
\sum_{y<p\leq x}|r_p(x)|\geq x(1-\ln 2+o(1))\sum_{y<p\leq x}\frac{1}{p}.
$$
For $y=x^{1-\varepsilon}$ we would get $\gg x$. Also, a trivial upper bound for $r_p(x)$ is $\frac{x}{p}$, so for $y\geq \sqrt{x}$ there is no hope for an upper bound of non-trivial order.
Hildebrand, in the paper you cite, discusses briefly its origins (last paragraph of the introduction).
This identity seems to have first appeared in Delange's paper in 1961, titled "Sur les fonctions arithmétiques multiplicatives" (Ann. Sci. École Norm. Sup. (3) 78 1961 273–304). It has been the starting point of many investigations in multiplicative number theory, old and new. It allows one to study the mean value of a function through its mean value on primes, and to reduce questions about multiplicative functions to question on integral equations.
I'll start by summarizing the general formula.
Given any multiplicative function $f$, subject only to $f(1)=1$, one can define an arithmetic function $\Lambda_f$ uniquely through the relation
$$f * \Lambda_f = f \cdot \log.$$
This relation can also be encoded in terms of (formal) Dirichler series:
$$ -\frac{F'(s)}{F(s)} = \sum_{n \ge 1} \frac{\Lambda_f(n)}{n^s} \qquad \left(F(s) =\sum_{n \ge 1} \frac{f(n)}{n^s}\right).$$
This later definition also shows $\Lambda_f$ is supported on prime powers. For any prime $p$ we have the recursive relation
$$\sum_{j=0}^{k-1} f(p^j)\Lambda_f(p^{k-j}) = f(p^k)\log(p^k), \qquad k \ge 1.$$
If $f$ is completely multiplicative, as in your question, one can verify $\Lambda_f(n)=\Lambda(n)f(n)$.
On the one hand,
$$\sum_{n \le x} f(n)\log n = \sum_{n \le x} (f*\Lambda_f)(n) = \sum_{p^m \le x} \Lambda_f(p^m) \mathcal{M}(x/p^m)$$
where I used your notation $\mathcal{M}(x) = \sum_{n \le x} f(n)$. On the other hand, Abel summation shows
$$\sum_{n \le x} f(n)\log n=\mathcal{M}(x) \log x - \int_{1}^{x} \frac{\mathcal{M}(t)}{t}dt.$$
Equating these two identities gives
$$(\star)\, \mathcal{M}(x)\log x = \int_{1}^{x} \frac{\mathcal{M}(t)}{t}dt + \sum_{n \le x} \Lambda_f(n) \mathcal{M}\left(\frac{x}{n}\right).$$
In the aforementioned paper of Delange, you will find the definition of $\Lambda_f$ in p. 277, equation (2) (where he uses the notation $c_j^{(p)}(f)$ for $\Lambda_f(p^j)/\log p$). In that paper, Delange considers $1$-bounded (possibly complex-valued) multiplicative functions, and proves in particular that if $\sum_{p} (1-f(p))/p$ converges then $f$ has an (explicit) mean value, and conversely, if $f$ has a non-zero mean value then $\sum_{p} (1-f(p))/p$ converges and $f(2^k)\neq -1$ for some $k\ge 1$.
The definition of $\Lambda_f$ is already useful in itself. Halász has a famous inequality holding for $1$-bounded multiplicative functions. It turns out that the correct framework for generalizing it to divisor-bounded multiplicative functions involves $\Lambda_f$: let $C(\kappa)$ be the set of functions with $|\Lambda_f(n)|\le \kappa \Lambda(n)$. Granville, Harper and Soundararajan, in "A new proof of Halász’s theorem, and its consequences" (Compos. Math. 155, No. 1, 126-163 (2019)), among other things, generalize the inequality to this class. See also their related paper "Mean values of multiplicative functions over function fields" (Res. Number Theory 1, Paper No. 25, 18 p. (2015)).
As a general principle, $(\star)$ allows one to study the mean value of a function through its mean value on primes. Letting
$$\sigma_f(u) := y^{-u} \sum_{n \le y^u} f(n)$$
and
$$\chi_f(u):= \left(\sum_{n \le y^u} \Lambda(n)\right)^{-1} \sum_{n \le y^u} \Lambda_f(n) \approx y^{-u} \sum_{p \le y^u} f(p)\log p,$$
then $(\star)$ can be used to show that $\sigma_f$ and $\chi_f$ approximately satisfy the integral equation
$$(\star \star) \, u\sigma(u) = \sigma*\chi(u):=\int_{0}^{u}\sigma(u-t)\chi(t)dt.$$
This idea has been used successfully by Wirsing in "Das asymptotische Verhalten von Summen über multiplikative Funktionen. II" (Acta Math. Acad. Sci. Hung. 18, 411-467 (1967)), where he proves (among other things) an asymptotic formula for $\sum_{n \le x}f(n)$ under very general conditions.
The connection between $(\star)$ and $(\star \star)$ has been used many times since. See especially "The spectrum of multiplicative functions" by Granville and Soundararajan (Ann. Math. (2) 153, No. 2, 407-470 (2001)), where in Proposition 1 and its converse they show in rather wide generality a correspondence between $(\sigma_f,\chi_f)$ and
the solutions $(\sigma,\chi)$ to $(\star \star)$ (in both directions). (See their result for accurate formulations.)
Another application of $(\star \star)$ appears in "The Elliott–Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture" by Tao (Algebra Number Theory 9, No. 4, 1005-1034 (2015)).
Best Answer
Yes, there are several results answering the question you ask about, in great generality.
Linnik, in his monograph on the dispersion method, proved an analogue of (2) for the $k$-th divisor function. Concretely, $$\sum_{\substack{n \le x \\ n \equiv a \bmod q}} d_k(n) \ll \frac{x}{q} \left(\frac{\phi^{r-1}(q)}{q^{r-1}}\log x \right)$$ uniformly for $q \le x^{1-\varepsilon}$.
This was generalized by Peter Shiu (a student of Halberstam) in his Crelle paper ''A Brun-Titchmarsh theorem for multiplicative functions'' (J. Reine Angew. Math. 313 (1980), 161–170) to a wide class of multiplicative functions. Here the bound takes the form $$\sum_{\substack{n \le x \\ n \equiv a \bmod q}} f(n) \ll \frac{x}{q} \frac{1}{\log x}\exp\left( \sum_{\substack{p \le x \\ p \nmid q}} f(p) \right)$$ uniformly for $q\le x^{1-\varepsilon}$, with an implied constant depending only on $f$ and $\varepsilon$. In fact, Shiu's result also allows you to add the additional restriction $n \in [x-y,x]$. Here $\frac{1}{\log x}\exp\left( \sum_{\substack{p \le x \\ p \nmid q}} f(p) \right)$ arises as the mean value of $\alpha$ over integers coprime to $q$ (up to a multiplicative constant bounded away from $0$ and $\infty$), by e.g. applying Wirsing's theorem to $f(n) \cdot 1_{(n,q)=1}$.
A further generalization was given by M. Nair and G. Tenenbaum in ''Short sums of certain arithmetic functions'' (Acta Math. 180 (1998), no. 1, 119–144), building on earlier work of Nair. In particular, they show that 'multiplicative' can be replaced by (a weak notion of) 'submultiplicaitve', dependence of the constant on $f$ can be made more specific, and the growth condition imposed by Shiu (which I did not mention) can be considerably weakened. Moreover, one can obtain similar bounds for (sub)multiplicative functions in several variables evaluated on polynomial arguments. This generalization leads to new results. A. Hildebrand, in his MathSciNet review calls this generalization 'far-reaching' and 'definitive'.
As far as I know, no author has tried to optimize the constant in the estimate (2) in the multiplicative setting.