How to interpret “confidence intervals” in observational studies

This question complements the one in the thread Random sampling versus random allocation/randomization- implications for p-value interpretation. Given that observational studies involve neither random sampling nor random allocation, why are they riddled with “95% confidence intervals”?

Sampling Error

Part 1c of Sampling and Confidence Intervals by Dr. Alvin Ang

Sample Size Planning for Statistical Power and Accuracy in Parameter Estimation | Annual Reviews

This review examines recent advances in sample size planning, not only from the perspective of an individual researcher, but also with regard to the goal of developing cumulative knowledge. Psychologists have traditionally thought of sample size planning in terms of power analysis. Although we review recent advances in power analysis, our main focus is the desirability of achieving accurate parameter estimates, either instead of or in addition to obtaining sufficient power. Accuracy in parameter estimation (AIPE) has taken on increasing importance in light of recent emphasis on effect size estimation and formation of confidence intervals. The review provides an overview of the logic behind sample size planning for AIPE and summarizes recent advances in implementing this approach in designs commonly used in psychological research.

Uncertainty Estimation with Conformal Prediction – Michael Clark

More options for uncertainty estimation

What fraction of repeat experiments will have an effect size within the 95% confidence interval of the first experiment?

Let's stick to an ideal situation with random sampling, Gaussian populations, equal variances, no P-hacking, etc. Step 1. You run an experiment say comparing two sample means, and compute a 95%

Understanding Precision-Based Sample Size Calculations | UVA Library