Suitable sample data set to test machine learning algorithms
The toy examples or the common datasets you are talking about are so because they are simple to visualise it and work with. Their simplicity helps the beginner to train simple models which don't require much compute. The simplicity in the structure of the dataset allows visualisation of the data on lower dimensions.

Reason for using them as test datasets is that they provide us a quick sanity check to see if the algorithm performs or not. The link you provided is specifically for clustering problems. So, datasets which can be easily visualised on 2D plane would be a simple dataset to check the performance of the algorithm via inspection. Had it been a complex datasets like a dataset of human faces, it would be difficult to evaluate the performance of the model through visualisation and inspection.

Some examples for such datasets:

MNIST dataset - collection of handwritten digits used to train classification network to identify the class of digit during test time.

Cifar-10 : collection of RGB images of 10 classes of objects in real world (e.g cars and birds).

Cifar-100: upgrade of Cifar-10. Contains images from 100 classes