If I have two process $X_t$ with rate λ and $Y_t$ with rate μ and I merge them I get an overall rate of λ + μ.
Is this the same logic that would follow if I have a process of $2X_t + 3Y_t$ to get 2λ + 3μ.I feel like it wouldn't be correct but I'm not sure what would be the correct answer then.
If by "rate $\lambda$" you mean a process $X_t$ such that $\mathbb E[X_t] = \lambda t$, then this is just linearity of expectation: $$\mathbb E[2 X_t + 3 Y_t] = 2 \mathbb E[X_t] + 3 \mathbb E[Y_t] = (2\lambda + 3 \mu) t$$
On the other hand, $2 X_t + 3 Y_t$ is certainly not a Poisson process (e.g. this never takes the value $1$), while if $X_t$ and $Y_t$ are independent Poisson processes, $X_t + Y_t$ is a Poisson process.