Clustering uncertain data with independent uncertainty per dimension
Model the measurement process

For each dimension you could attempt to model the distribution of the measurement process variance. Perhaps every dimension has the same measurement process or perhaps it is distinct. Combine all measurements from all dimensions if the process is the same.

If you have enough data then you can build an empirical model of the distribution of errors using perhaps a kernel based estimator.

Otherwise you might choose to use a Gaussian distribution formed from the mean value of each measurement and an overall measure of the variance of the measurement process. Then you replace observations you have with a sample drawn from a  distribution with the observation mean and overall measurement process variance.