I think I got the proof that no such real polynomial with degree $ \geq 6$ exists.
Let $n \geq 6$
Suppose for contradiction that $z_1,\ldots,z_n \in \mathbb R-\{0\}^n$ are such that $(X-z_1)...(X-z_n)=X^n+\sum_{k=1}^{n-1}z_iX^{n-i}$
Then three useful identities appear $$\sum_{k=1}^{n}z_k=-z_1 \; \; \; \;(1)$$
$$\sum_{\large1\leq i<j \leq n}z_iz_j=z_2 \; \; \; \;(2)$$
$$\prod_{k=1}^n z_k=(-1)^n z_n \; \; \; \;(3)$$
Since $$(\sum_{k=1}^{n}z_k)^2=\sum_{k=1}^{n}z_k^2+2\sum_{\large1\leq i<j \leq n}z_iz_j$$it follows that$$z_1^2=2z_2+\sum_{k=1}^{n}z_k^2$$
Hence $$0< \sum_{k=2}^{n}z_k^2=-2z_2 \; \; \; \;(4)$$ and $$0<\sum_{k=3}^{n}z_k^2=1-(z_2+1)^2 \; \; \; \;(5) $$
$(4)$ and $(5)$ imply $$\; \; \; \;-2<z_2<0 \; \; \; \;(6)$$
thus $(6)$ and $(4)$ imply $$0<\sum_{k=2}^{n}z_k^2 < 4 \; \; \; \; (7)$$
Also $(6)$ and $(5)$ imply $$0<\sum_{k=3}^{n}z_k^2 \leq 1 \; \; \; \; (8)$$
By AM-GM, $$\left(|z_3|^2\ldots|z_{n-1}|^2 \right)^{1/(n-3)} \leq \frac{1}{n-3}\sum_{k=3}^{n-1}z_k^2 \leq \frac{1}{n-3}\sum_{k=3}^{n}z_k^2$$
Hence
$$|z_3|^2\ldots|z_{n-1}|^2 \leq \left(\frac{1}{n-3}\sum_{k=3}^{n}z_k^2\right)^{n-3} $$
Squaring, $$|z_3|\ldots|z_{n-1}| \leq \left(\frac{1}{n-3}\sum_{k=3}^{n}z_k^2\right)^{\large \frac{n-3}{2}} \leq_{ \large (8)} \dfrac{1}{{(n-3)}^{(n-3)/2}} \; \; \; \; (9)$$
By triangle inequality $(1)$, and Cauchy-Schwarz
$$2|z_1| \leq \sum_{k=2}^{n}|z_k| \leq \sqrt{n-1} \sqrt{\sum_{k=2}^{n}z_k^2} $$
Hence by $(7)$,
$$|z_1| \leq \sqrt{n-1} \; \; \; \; (10)$$
Rewriting $(6)$ as $$|z_2|\lt2 \; \; \; \; (11) $$
Recalling $(3)$ (with $z_n$ cancelled from both sides) and putting together $(9)$, $(10)$ and $(11)$, we have
$$1=|z_1||z_2||z_3|\cdots|z_{n-1}| < \dfrac{ 2\sqrt{n-1}}{{(n-3)}^{(n-3)/2}}$$
This inequality fails for $n\geq 6$.
Contradiction.
I can't prove anything for $n=5$ so maybe the conjecture doesn't hold.
Suppose $r$ is a root of the polynomial. Then $p(r^2) = p(r) p(r+1) = 0$ and $p((r-1)^2) = p(r-1) p(r) = 0$, so $r^2$ and $(r-1)^2$ are
roots as well.
In order to avoid generating an infinite sequence of
distinct roots, each the square of the previous one, we need $|r| = 0$ or $|r| = 1$. Similarly we need $|r-1| = 0$ or $1$, and
$|r^2-1| = 0$ or $1$. It's not hard to show that the only possible roots are $0$ and $1$.
EDIT: So let $p(x) = a x^m (x-1)^n$.
By considering the leading coefficient of $p(x^2) - p(x)p(x+1)$, we find that $a = 1$. The zero of $p(x^2)$ at $0$ has order $2m$, while the zero of $p(x) p(x+1)$ there
has order $m+n$, so $m = n$.
And finally, we find that $p(x) = x^m (x-1)^m$ does satisfy the equation.
Best Answer
Case $n>1$
As you noticed $P(0)=0$. Using this fact and evaluate the equality in $x=n$ you have: \begin{gather} nP(n-n) = (n-1)P(n)\\ 0 = P(n) \end{gather}
This procedure suggests (in some sense) the following statement:
Infact, evaluating the eqaulity in $(k+1)n$ knowing that $P(kn)=0$ we have: \begin{gather} (k+1)n P((k+1)n-n)) = ((k+1)n-1)P((k+1)n)\\ 0 = P((k+1)n) \end{gather}
Thanks to this fact, you have that the set $\{0,n,2n,3n, 4n,...\} = \{kn\}_{k\in \mathbb N}$ is a set of roots of $P$. Since it is infinity, $P(x)=0$.
Case $n=1$
Again we have $P(0)=0$ so $P(x)=xQ(x)$ for a certain polynomial $Q(x)$. Sobstituting this equality in the equality of the text we have: \begin{gather} x(x-1)Q(x-1)=x(x-1)Q(x)\\ Q(x-1)=Q(x) \end{gather} And this implies that $Q(x)=c$ with $c\in \mathbb R$. Then the polynomial $P(x)$ is necessarly of the form $P(x)=cx$ for some $c\in \mathbb R$ and every polynomial of this form works.
Edit: In the case $1$ we have to take linear increment and not exponential.