You have to realize that you are asking a really vague question. Specifically, you haven't defined "similar (nice) recurrence relation" and "some relationship between various points on the curve".
When you say similar recurrence relation, I believe the next step to generalizing Chebyshev polynomials is to consider orthogonal polynomials, which satisfy a recurrence relation, but at the same time they come from certain differential equations which sometimes allow us to express them using Rodrigues' formula. For Chebyshev polynomials, this connects their recurrence relation $$T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)$$ to the formula $$T_n(x)=\frac{\Gamma(1/2)\sqrt{1-x^2}}{(-2)^n \Gamma(n+1/2)}\frac{d^n}{dx^n}\left([1-x^2]^{n-1/2}\right).$$ Similar formulas apply to most families of orthogonal polynomials, however the formula $$T_n(x)=cos(n \arccos(x))$$ is a bit special and characteristic of Chebyshev polynomials. In fact if you had a family of polynomials $\{P_k\}_{k\geq 1}$ so that you could express them as $$P_n(x)=f(nf^{-1}(x))$$ for some function $f$, then they would trivially satisfy $P_n(P_m(x))=P_m(P_n(x))=P_{mn}(x)$ and by a result of Block and Thielman, these polynomials must coincide with a linear transformation of either the power sequence $\{x^k\}$ or the Chebyshev polynomials $\{T_k\}$.
Another perspective which sheds a little more light on the geometric properties of Chebyshev polynomials is Dessin d'enfant. The Belyi functions corresponding to trees are the Shabat polynomials which are generalizations of Chebyshev polynomials. I believe this is the right context to observe the symmetries of functional composition of Chebyshev functions. See these articles.