Is there any guarantee that there will be only one point on the isotherm curve of a system that has a certain Y coord? When reading this part of the "heat and thermodynamics" book of Zemansky,

is there any guarantee that there will be only ONE intersection of the $Y=Y_1$ and the isotherm line?
The alternative of that would be to prove that all isotherms are one-to-one functions of $X$. Why then?
There must be. Because otherwise the concept of temperature would be self-disproving.

Link to the book. Not sure about the copy right. To download, press the green button in the dark part if you are confused. That part is in page 9.

Will the situation differ if we restrict our selves 1. to fluids only and 2. to ideal gasses?
 A: At the level of abstraction at which this is stated in the book, there certainly exists no such guarantee. 
It is only assumed (in 1.5 of your book) that a system is described by two coordinates $X$ and $Y$. Within this space, an isotherm is the locus of all states that are in thermal equilibrium (1.6). By the zeroth law of thermodynamics, thermal equilibrium is an equivalence relation, so that these isotherms partition the full state space. It is stated that no assumption is made on the nature of these isotherms, but that in practice they are essentially continuous curves.
In 1.7 these isotherms are used to define temperature scales, and here the assumption that they are curves becomes important. Now we have a 2-dimensional state space that is partitioned into a family of curves, so every state belongs to a unique isotherm. If this partition is more or less well-behaved, the set of isotherms can be parametrized by a single curve itself (e.g. a curve in the 2D space that intersects each isotherm exactly once). An example would be your line $Y = Y_1$ in figure 1-4.
In general, such a line will not work. Even in figure 1-4, a change of coordinates corresponding to a rotation could make that horizontal lines intersect some isotherms twice, and others not at all. 
More generally, under the limited set of assumptions made so far, it could happen that no such curve can exist, e.g. if you have closed isotherms (in that case there has to be a one-point isotherm by the way, something that is certainly physically acceptable). If there are two such points, the curve has to start on one and finish on the other (and even then it may not be able to intersect all isotherms once). Note that this case would allow you to define an absolute minimum and an absolute maximum. If there are more than two, you certainly cannot define a temperature.
All this shows that to be able to define temperature from such a system (called a thermometer) some more assumptions are implicitly made. Finally the assumption that the $X$ coordinate can be used as a thermometric property doesn't directly constrain the systems that can be used as thermometers, but it does say something about their coordinatization.
