I'll wade in cautiously here. buried deep in the concept and the calculations of the constants is the core concept that also results in a 'confidence interval'. However, as Steve pointed out, the control chart limits are not confidence intervals. Confidence intervals and control limits can be thought of as cousins. and there is certainly no attempt or ability or need to have a precise probability of any particular result with a control limit as Steve also pointed out.

certainly any particular result has some large or small probability of occurring If the process hasn't changed. The same can be said when comparing two point estimates to each other...but confidence intervals are intended for comparing a very small number of point estimates, typically not differentiated by time but be a different source of change like different vendors, equipment, materials or processes. control limits are used to compare often large numbers of samples that differentiate themselves in time sequence. These different comparisons while similar in some aspects are really very different types of comparisons...