Clebsch–Gordan decomposition formula for algebraic groups $\DeclareMathOperator\SL{SL}$There is a well-known Clebsch–Gordan decomposition formula for irreducible representations of $\SL_2$. If $V_n$ denotes the unique $n+1$-dimensional irreducible representation of $\SL_2$ we have that for $n \geq m$:
$$ V_n \otimes V_m = V_{n+m} \oplus V_{n+m-2} \oplus \dots \oplus V_{n-m}. $$
I wonder if similar formulas exist for any other algebraic groups, for instance, $\SL_3$ or the symmetric group $S_n$.
Böhning and Bothmer - A Clebsch–Gordan formula for $\SL_3(\mathbb C)$ and applications to rationality uses an algorithm with Young tableaux to decompose the tensor products of two irreducible representations of $\SL_3$. More about this algorithm can be found in chapter 12 of Georgi - Lie algebras in particle physics. For $S_n$, I found Schindler - The decomposition of the tensor product of representations of the symmetric group, but no explicit formula is given.
 A: Let $G$ denote the group, and suppose one has an enumeration of its irreducible representations $V_{\lambda}$ by some combinatorial objects $\lambda$, like integers for $SU_2$ (or finite dimensional non-unitary representations of $SL_2$) or integer partitions, etc. There are two different Clebsch-Gordan (CG) problems which the OP seems to conflate.
CG1) The numerical CG problem:
It is to figure out the multiplicities $m(\lambda,\mu;\nu)\in\mathbb{N}$ in the general decomposition into irreducibles of tensor products of two irreducibles:
$$
V_{\lambda}\otimes V_{\mu}=\bigoplus_{\nu}V_{\nu}^{\oplus m(\lambda,\mu;\nu)}\ .
$$
CG2) The explicit CG problem: It is to realize the above decomposition with explicit intertwiners, namely, to write a decomposition of the identity operator $I_{V_{\lambda}\otimes V_{\mu}}$ on $V_{\lambda}\otimes V_{\mu}$ in the form:
$$
I_{V_{\lambda}\otimes V_{\mu}}=\sum_{\nu}\sum_{j=1}^{m(\lambda,\mu;\nu)}
\iota_{\lambda,\mu,\nu,j}\circ\pi_{\lambda,\mu,\nu,j}
$$
where $\pi_{\lambda,\mu,\nu,j}\in {\rm Hom}_G(V_{\lambda}\otimes V_{\mu},V_{\nu})$ and $\iota_{\lambda,\mu,\nu,j}\in {\rm Hom}_G(V_{\nu},V_{\lambda}\otimes V_{\mu})$ are explicit $G$-equivariant maps, i.e., intertwiners.
Note that to be able to even ask the question, a prerequisite is to solve
CG0) The parametrization of irreducibles: Namely, understanding the list of irreducibles, and having a parametrization $\lambda\mapsto V_{\lambda}$.
For $SU_2$, $SL_2$ all these problems were solved by Paul Gordan and Alfred Clebsch in the mid 19-th century,
see Section 2 of my article:

*

*A. Abdesselam, On the volume conjecture for classical spin networks. J. Knot Theory Ramifications
21 (2012), no. 3, 1250022, 62 pp.

Problem CG1 for $SU_n$, $SL_n$, $GL_n$ has been solved, and the multiplicities are the so-called Littlewood-Richardson coefficients.
For $S_n$, CG1 is much more difficult. The multiplicities are the Kronecker coefficients, and there is no satisfactory combinatorial description for them.
The recent article by Böhning and Graf von Bothmer does not just solve CG1 for $SU_3$, $SL_3$ (that's known from a long time ago), but rather problem CG2 for these groups. The case of $SL_n$, $GL_n$ is still open. When $\mu$ is the fundamental representation (adding a single box), there are some results, see

*

*M. Hunziker, J. A. Miller, and M. Sepanski. Explicit Pieri Inclusions. Electronic J. of Combinatorics 28 (2021), no. 3, P3.49.

and references therein (in particular some older work by Peter Olver).
As for CG0 in the case or $SL_n$, it was solved by Alfred Clebsch in the 1870's, and later by Deruyts, and then Schur, see:

*

*D. E. Littlewood, Invariant theory, tensors and group characters, Philo. Trans. Royal Soc. London Ser. A 239 (1944) 305–365.

For $S_n$, CG0 was solved by Alfred Young and later by Specht. A good account is
in the lectures by Adriano Grasia Alfred Young’s construction
of the irreducible representations of $S_n$.
Finally, the only instance of CG2 related work for $S_n$ that I am aware of is:

*

*J. Chipalkatti and T. Mohammed,
Standard tableaux and Kronecker projections of Specht modules.
International Elec. J. Algebra 10 (2011), no. 10, 123-150.

