We work in Euclidean space-time.
OP is trying to compare the two prescriptions
$$\tag{A}
\int_{\mathbb R^{na}} f(p_1,\dots,p_a)\ \mathrm d^np_1\cdots\mathrm d^np_a\ \to\ \mu^{a\epsilon}\int_{\mathbb R^{da}} f(p_1,\dots,p_a)\ \mathrm d^dp_1\cdots\mathrm d^dp_a
$$
and
$$\tag{B}
\int_{\mathbb R^{na}} f(p_1,\dots,p_a)\ \mathrm d^np_1\cdots\mathrm d^np_a\ \to\ \tilde\mu^{a\epsilon}\int_{\mathbb R^{na}} \frac{f(p_1,\dots,p_a)}{|p_1|^\epsilon\cdots|p_a|^\epsilon}\, \mathrm d^np_1\cdots\mathrm d^np_a
$$
where $(n,a)\in\mathbb N^2$ is a pair of integers; $d\in\mathbb C$ is a complex parameter; $\epsilon:=n-d$; and $(\mu,\tilde\mu)\in\mathbb R^2$ is a pair of real parameters (mass scales).
We stress that $(\mathrm A)$ is the standard dimensional regularisation prescription. The case $(\mathrm B)$ is quite similar to Speer's analytic regularisation, but it is not equivalent.
If $f$ depends on the momenta only through their square,
$$
f(p_1,\dots,p_a)=f(p_1^2,\dots,p_a^2)
$$
then it's easy to check that the prescriptions above are equivalent to
$$\tag{A}
\int_{\mathbb R^{na}} f(p_1,\dots,p_a)\ \mathrm d^np_1\cdots\mathrm d^np_a\ \to\ \mu^{a\epsilon}\Omega_d^a\int_{\mathbb R^{a}} p_1^{d-1}\cdots p_a^{d-1}f(p_1,\dots,p_a)\ \mathrm dp_1\cdots\mathrm dp_a
$$
and
$$\tag{B}
\int_{\mathbb R^{na}} f(p_1,\dots,p_a)\ \mathrm d^np_1\cdots\mathrm d^np_a\ \to\ \tilde\mu^{a\epsilon}\Omega_n^a\int_{\mathbb R^{a}} p_1^{n-1-\epsilon}\cdots p_a^{n-1-\epsilon}f(p_1,\dots,p_a)\, \mathrm dp_1\cdots\mathrm dp_a
$$
and we see that the two prescriptions are mapped into each other under
$$
\tilde\mu^\epsilon\to \frac{\Omega_d}{\Omega_n}\mu^\epsilon
$$
As $\mu,\tilde\mu$ are arbitrary, the two prescriptions are equivalent.
If $f$ contains a non-trivial spinorial structure, the equivalence is still correct if we agree that traces in $d$ dimensions satisfy
$$
\text{tr}(1)=2^{\lfloor n/2\rfloor}
$$
for all $d\in\mathbb C$.
On the other hand, if $f$ contains non-trivial Lorentz structure (i.e., if it's not a Lorentz scalar or it depends on momenta not only through their squares, but also on the scalar products $p_i\cdot p_j$), then the correspondence breaks down. A simple way to see this is that, according to the standard dimensional regularisation prescription we have
$$
p^\mu p^\nu\to\frac{\delta^{\mu\nu}}{d}p^2
$$
while in the modified prescription we have
$$
p^\mu p^\nu\to\frac{\delta^{\mu\nu}}{n}p^2
$$
These are not equivalent, and cannot be made equivalent by modifying either prescription (we cannot define $p^\mu p^\nu$ to be $\frac{\delta^{\mu\nu}}{n}p^2$ in $d$ dimensions, because algebraic manipulations such as contractions and shifts in the integration variable would lead to inconsistencies). While it's true that the divergent part is the same in both approaches, the finite part need not.
More generally, the two prescriptions differ by a rational function of the form
$$
\frac{P(n)}{P(d)}
$$
where $P$ is a polynomial. This polynomial is not in general the same for all Feynman diagrams that contribute to a certain order in perturbation theory (see e.g. the case of scalar QED, cf. ref1, chapter 65). Therefore, one does not expect a cancellation of the $\xi$-dependent terms, and the Ward identity is violated. This cancellation holds in the case of dimensional regularisation, so the two prescriptions are not equivalent.
What's more, not only is the gauge symmetry anomalous in OP's scheme, the axial symmetry is not. Indeed, if $n$ is even, we can define the axial matrix $\gamma_5$. This matrix is traceless $\text{tr}(\gamma_5)=0$ for any number of spacetime dimensions, so the divergence of the axial current vanishes (in dimensional regularisation, this argument fails because $\gamma_5$ is ill-defined for complex $d$; but in OP's prescription the number of spacetime dimensions $n$ is fixed). As the axial anomaly does not vanish for any (even) $n$ (cf. this PSE post), the OP's regularisation is not equivalent to dimensional regularisation.
Finally, it bears mentioning that there is an intermediate prescription,
$$\tag{C}
\int_{\mathbb R^{na}} f(p_1,\dots,p_a)\ \mathrm d^np_1\cdots\mathrm d^np_a\ \to\ \hat\mu^{a\epsilon}\frac{\Omega^a_d}{\Omega_n^a}\int_{\mathbb R^{da}} f(p_1,\dots,p_a)\ \mathrm d^dp_1\cdots\mathrm d^dp_a
$$
which is equivalent to dimensional regularisation. This prescription leaves the angular measure as that of $n$ instead of $d$, so it is rather close to what OP was after. But the integrand is evaluated at complex $d$ rather than at fixed $n$, so it is the same thing as dimensional regularisation (or rather, it is mapped into it under the aforementioned rescaling of the mass scale).
References.
- Srednicki's QFT.
