In this paper, we first consider the normalized solutions for the following Schrödinger system \(\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )_p^s u+\mathcal {V}(\varepsilon x)|u|^{p-2}u=\lambda _1 |u|^{p-2}u+\mu _1|u|^{{q_1}-2}u+\beta r_1|u|^{r_1-2}u|v|^{r_2} \quad \text{ in }\ \mathbb {R}^N,\\ (-\Delta )_p^s v+\mathcal {W}(\varepsilon x)|v|^{p-2}v=\lambda _2 |v|^{p-2}v+\mu _2|v|^{{q_2}-2}v+\beta r_2|u|^{r_1}|v|^{r_2-2}v \quad \text{ in }\ \mathbb {R}^N,\\ \end{array}\right. \end{aligned}\) under the constraints \(\int _{\mathbb {R}^N}|u|^p\text {d}x=a^p \ \ \text{ and } \ \ \quad \int _{\mathbb {R}^N}|v|^p\text {d}x=b^p,\) where \((-\Delta )_{p}^s\) is the fractional p-Laplace operator, \(s\in (0,1)\) , \(r_1,r_2>1\) , \({q_1},{q_2},r_1+r_2\in (p,p+\frac{p^2s}{N}), \varepsilon >0\) is a parameter, \(\mu _1,\mu _2,a,b>0,\) \(\beta >0\) are prescribed, \(\lambda _1,\lambda _2\) are the Lagrange multipliers to be determined, and the potential functions \(\mathcal {V},\mathcal {W}:\mathbb {R}^N\rightarrow \mathbb {R}\) both have global minimum. With the aid of Lusternik–Schnirelmann category theory and variational method, we obtain the multiplicity and concentration phenomena of normalized solutions for the above system in \(L^p\) -subcritical case. Moreover, we also use Morse theory to get the existence of multiple normalized solutions for Schrödinger system. To our best knowledge, it is the first time that the Lusternik–Schnirelmann category theory and Morse theory are applied to study the normalized solutions of Schrödinger system. Our results are even new in the case p-Laplace as \(s\rightarrow 1^{-}\) and fractional Laplace as \(p=2.\)
