Interesting question! The concept of statistical variable can be thought of being a characteristic/number/quantity whose 'values' change across the items for which this characteristic is assessed. For example, weight changes from one person to another, so we can think of weight as being a variable.
When we collect the values of a statistical variable such as weight, we can record them in different ways. For example, we could record the values as 49.4 kg, 68.9 kg, 72.3 kg, etc.
Often, we record the values in one way but may choose to analyse them in a different way. As an example, we might choose to convert the values of weight recorded above to <50 kg versus 50+ kg.
So what really matters at the end is how we treat a statistical variable at the analysis phase (which is in turns influenced by how we recorded its original values, what kind of research questions we are interested in, etc.)
A statistical variable is said to be random when its values are assessed across items which are selected from a larger set of items using a random mechanism. As an example, if we plan on measuring weight for 100 patients selected at random from a local hospital, then weight is a random variable. Or if we measure weight for patients assigned at random to one of two diet regimens and measure their weight, weight is a random variable. The values of weight are unknown to us when we design the study but are expected to vary at random across patients. (A variable that is not random is called non-random.)
Whether a statistical variable is random or non-random, its values can be broadly categorized as qualitative or quantitative.
You may encounter definitions such as "A variable is qualitative if it can not be represented by a number". (https://internal.ncl.ac.uk/ask/numeracy-maths-statistics/statistics/descriptive-statistics/variables.html) This definition is not correct: you can have qualitative variables whose values are coded numerically (e.g., 1, 2, 3) such that the codes represent labels (e.g., 1 = low, 2 = medium, 3 = high).
Basically, a qualitative variable is one whose values represent distinct, non-overlapping labels with a specific meaning. The labels can be things like 'low, medium, high' or 'male, female, other' and they can be coded numerically. A qualitative variable can be further categorized into nominal (labels have no intrinsic ordering) or ordinal (labels have an intrinsic ordering).
On the other hand, a quantitative variable is one whose values are numbers (though the numbers are not codes for some underlying labels) which refer to counts, quantities, amounts, ratios, proportions, percentages, etc.
You will see quantitative variables being categorized into continuous or discrete. Discrete variables can take on (i) a finite number of values or (ii) an infinite number of values. Continuous variables can take on an infinite number of values.
Getting back to statistical variables that are random, then these can be nominal random variables, ordinal random variables, continuous random variables or discrete random variables. (Same for non-random variables!)