Setting 'Show Feature Count' adds a number of features in layer's name, but doesn't actually change its name, in a sense that its name can still be got from the 'name()' method and the result would not include the feature count within brackets.

Is it possible to show custom properties in a same way with pyqgis or ones already available in information panel?

For instance, setting CRS to show as: {layer_name} [EPSG:4326]?