It is okay to reuse or “recycle” symbols - I sometimes joke that this is actually good practice since it is “environmentally friendly” and saves mental overhead. However, you must make sure that you do not use the same symbol to denote two different quantities within the same proof or calculation. That is a mistake and invites potential confusion or misunderstanding, even if in some cases the reader can perform disambiguation of the symbol based on context and figure out the correct interpretation.
A way to think about this is that each notation has a scope: it is created, used, and then discarded, often within the space of a single proof or even a couple of paragraphs. Some notations are local and live for a short time; others are global and retain the same meaning throughout the paper or thesis. Overlapping notations are fine if the scopes of the notations do not overlap. This is analogous to the use of variable names in a computer program. (Or to use a physics metaphor, notations can overlap in alphabet space, but not in spacetime - their world lines should not intersect...)
Finally, a list of notations should only contain notations that exist for a significant length of time (say for a whole chapter of the thesis), not “temporary” or ad hoc notations that are used in a short argument. That’s what matters, not whether the quantity represented by the notation is integer-valued. The main consideration is whether the notation is used long enough that a reader might encounter it far from where it is defined and have difficulty figuring out its definition.
It is also a good idea for the notation table to include columns to specify the page number where each notation is defined and which chapters or sections it is used for, to make lookup easier.