1

The tongue is characteristic with yellow coating at the base, or it may be dry, smooth, glazed and cracked.

X is characteristic with y

I cannot get the concept of the bold parts.

2

This phrasing is sometimes used to mean that x is a defining feature or way to identify something (in this case, presumably a disease), with your y giving details. You could also think of this as:

Yellow coating at the base of the tongue is characteristic, or it (the tongue) may be dry, smooth, glazed and cracked.

It is quite common with medical descriptions for x to be some sort of diagnostic procedure:

The histology is characteristic, with...

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