Example 15 Find all non-zero real-coefficient polynomials $P(x)$ satisfying $P(x) \cdot P\left(2 x^{2}\right) \equiv P\left(2 x^{3}+x\right)(x \in \mathbf{R})$.
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Let the non-zero polynomial $P(x)=a_{n} x^{n}+\cdots+a_{1} x+a_{0}\left(a_{n} \neq 0\right)$ satisfy the conditions given in the problem. By comparing the coefficients of $x^{3 n}$ and $x^{0}$ in the original identity, we get $a_{n}^{2}=a_{n}, a_{0}^{2}=a_{0}$, thus $a_{n}=1$. If $a_{0}=0$, then $P(x)=x^{k} P_{1(x)}$, where $P_{1}(0) \neq 0, k \in \mathbf{N}$. From the identity $x^{k} P_{1}(x) \cdot\left(2 x^{2}\right)^{k} \cdot P_{1}\left(2 x^{2}\right) \equiv\left(2 x^{3}+\right.$ $x)^{k} \cdot P_{1}\left(2 x^{3}+x\right)$, we have $2^{k} x^{2 k} \cdot P_{1}(x) \cdot P_{1}\left(2 x^{2}\right) \equiv\left(2 x^{2}+1\right)^{k} \cdot P_{1}\left(2 x^{3}+x\right) \Rightarrow$ $P_{1}(0)=0$, which is a contradiction. Therefore, $a_{0}=1$.

Suppose $\alpha=r(\cos \varphi+\mathrm{i} \sin \varphi)$ is a root of the polynomial $P(x)$. Since $P\left(2 \alpha^{3}+\alpha\right)=P(\alpha) \cdot P\left(2 \alpha^{2}\right)$, it follows that $2 \alpha^{3}+\alpha$ is also a root.

Furthermore, $\left|2 \alpha^{3}+\alpha\right|=|\alpha| \cdot\left|2 \alpha^{2}+1\right| \geqslant|\alpha| \cdot\left(2|\alpha|^{2}-1\right)$. Therefore, if $|\alpha|=$ $r>1$, then $\left|2 \alpha^{3}+\alpha\right|>|\alpha|$, which implies that the non-zero polynomial $P(x)$ has infinitely many distinct roots: $\beta_{1}=\alpha, \beta_{j+1}=$ $2 \beta_{j}^{3}+\beta_{j}, j=1,2, \cdots$, which is impossible. Thus, the roots of the polynomial $P(x)$ have a modulus no greater than 1. However, by Vieta's formulas, the product of the roots is 1, so there are no roots with a modulus less than 1. Therefore, $r=1$, and from $1=\left|2 \alpha^{3}+\alpha\right|^{2}=\left|2 \alpha^{2}+1\right|^{2}=$ $4 \cos 2 \varphi+5$, we get $\varphi=\frac{\pi}{2}+m \pi, m \in \mathbf{Z}$, i.e., $\alpha= \pm \mathrm{i}$. Since $\beta_{j}$ is real, we have $P(x)=$ $\left(x^{2}+1\right)^{k}, k \in \mathbf{Z}^{+}$.
Finally, it can be verified that all polynomials of this form satisfy the problem's conditions.