tl;dr Sure we can call Gourgoulhon's $K$ "the duration of one period of an ideal (ticking) clock"; and the inverse, $1/K$, "the (ticking) rate". And we may think of distinct (ticking) clocks having unequal (ticking) rates -- e.g. an atomic clock vs. a mechanical wristwatch; perhaps we might even encounter some of them.
Preliminaries concerning terminology and notation
The OP question contained a quite selective quote of Éric Gourgoulhon's definition of an ideal clock, from "Special Relativity in General Frames: From Particles to Astrophysics" (2013), sect. 2.3.2, and I'd like first of all to fill in some gaps of terminology and notation according to my own preference:
According to definition steps (i) and (ii) of the quoted definition, a clock is a physical device, thought of as being approximated by a "(material) point", following a (certain segment of a) timelike world line; and any world line is consisting of events, of course.
At any one event, however, in general there may have been several distinct "(material) points" participating; and each of them may itself be thought of approximating one distinct clock. And some of those clocks may indicate a tick (and indeed each of them its very own distinguishable "kind of" tick) as they jointly participate in the specific one event under consideration; and others may not.
Each tick of any one clock should therefore not constute one whole event in which the clock participated (as Gourgoulhon prescribes), but instead only the part of the event which is attributable the one distinctive clock under consideration, "its indication at this event"; without the "event parts" ("indications") of any other clocks which happened to participate in this same event, and whose own indications at this event may or may not have been ticks as well.
Consequently, a clock does have a sequence of ticks which sample its world line, as stipulated in definition step (iii); but those ticks are not whole events, but certain conspicuous indications of this clock.
The notation should and can express these considerations:
plain letters such as $A$ for an identifiable physical device, or the correspondigly approximated material point;
the ordered set $\mathcal A$ for the tick indications which are stated and signalled by $A$;
a specific, one-to-one, consecutive enumeration of those ticks, as function $t_{\mathfrak A} : \mathcal A \longleftrightarrow \mathbb Z$.
A specific ticking clock $\mathfrak A$ is then characterized by all that, explicitly as a triple of symbols: $\mathfrak A \equiv (A, \mathcal A, t_{\mathfrak A})$.
The corresponding equivalent of Gourgoulhon's equation (2.11), (page 33), which can be considered the essence of the definition of an ideal clock and where the constant factor $K$ makes its first appearance, could consequently be written as
$$\tau [ \, (t_{\mathfrak A})^{(-1)}[ \, j \, ], (t_{\mathfrak A})^{(-1)}[ \, j + N \, ] \, ] = K \, N. \tag{1} $$
Or more straightforward for instance as
$$\tau [ \, A_{(j)}, A_{(j + N)} \, ] = K \, N. \tag{2} $$
On naming the constant factor $K$, and its inverse, $1/K$
Sure we can call Gourgoulhon's $K$ "the duration of one period of an ideal (ticking) clock"; and the inverse, $1/K$, "the (ticking) rate".
The quantity $\tau$, referring to a specific world line segment of a specific material point $A$, can also be called "$A$'s duration (from one of its indications, until another one)".
And for any two distinct (and distinctly enumerated) tick indications $A_{(p)}$ and $A_{(q)}$ of an ideal clock $\mathfrak A$ the ratio $$\frac{(q - p)}{\tau [ \, A_{(p)}, A_{(q)} \, ]} \equiv K_{\mathfrak A}$$ is an expression and the value of its ticking rate.
Comparison and possible inequality of the (ticking) rates of distinct (ideal) clocks
Of the separately constant ticking rates of two clocks, $\mathfrak A$ and $\mathfrak B$, their ratio can be expressed and evaluated as
$$ \frac{K_{\mathfrak A}}{K_{\mathfrak B}} = \left(\frac{(q - p)}{(v - u)}\right) \times \left( \frac{\tau [ \, B_{(u)}, B_{(v)} \, ]}{\tau [ \, A_{(p)}, A_{(q)} \, ]} \right), \tag{3} $$
for any two distinct (and distinctly enumerated) tick indications $A_{(p)}$ and $A_{(q)}$ of ideal clock $\mathfrak A$, and any two distinct (and distinctly enumerated) tick indications $B_{(u)}$ and $B_{(v)}$ of ideal clock $\mathfrak B$.
If this ratio evaluates to $1$ then clocks $\mathfrak A$ and $\mathfrak B$ are said to have ticked at equal rates; otherwise at unequal rates.
(How to measure the ratio between two durations itself, e.g. how to evaluate the right-hand term of expression (3), is of course a topic by itself, which is dealt with in the special and in the general theory of relativity).
