In the above case data is not [2]. 2 is the real value which you are trying to predict. How you normally proceed is you measure it say 5 times and you get [2.1, 1.99, 2.02, 1.95, 1.99]. And you say
data ~ Normal(2, 0.1)
That is where your first model looks weird to start with when you seem to have replaced your observations with the mean of your simulated data.
If uncertainty was described by an irregular distribution, then it could be, for example, multimodal, and there is no way of putting multimodal into “Y_obs” and describing it in terms of σ and μ.
Of course, if say the uncertainty due to measurement can not be parametrized by some parameters and a known distribution then you are pretty much out of luck with classical Bayesian approaches. In classical statistics this would be the difference between parametric (MLE etc) and non-parametric (emprical CDF etc) approaches.
What you are trying to achieve here is also outside the boundaries of classical Bayesian approaches but atleast there is one way to attempt to get to it, where as if you knew nothing about how this uncertainty is generated then there is nothing you can do I guess.
I ran your model, and it works, but then I adjusted it for my priors, noise, and observed data, and it does not converge, probably because the noise is too high.
Can you post a code that shows how you adjusted the parameters?