What are the ways available to figure out whether the data is sample data or population data apart from the client's information?
There is no way - the "population" of interest is part of the specification of the problem.
Statistical problems involving inference to a "population" require specification of the group of interest, about which we are making an inference. Only a proper specification of the problem ---in this case, from a briefing from the client--- can give you this. Of course, there may be situations where the client does not know how to specify their problem in a well-posed way, and in this case, part of the responsibility of the statistician is to elicit contextual information to assist the client to formulate a well-posed problem. In some cases, the source of existing sample data may also imply some natural suggestions about the "population" for which we can make a valid inference. (Generally, a random sample allows us to make an inference about characteristics of the corresponding sampling frame, which may be close to some population of common interest.) Sample data cannot formulate your statistical problem for you. The problem must arise from some objective or context.
As to whether data is "sample data" or "population data", that also depends on context, and specification of the group of interest. For example, suppose we consider data on the driving record (demerit points, fines, years with license, etc.) of a random sample of people with driver's licenses registered in a particular State. That data would be "sample data" from the associated sampling frame from which they were drawn ---i.e., all people who hold a driver's license registered in that State--- and the data of all people with a driver's license registered in that State would be the "population". However, that "population" can also be regarded as (non-randomised) "sample data" from the larger class of all people with driver's licenses registered anywhere in the country, which can in turn be considered as (non-randomised) "sample data" from the larger class of all people with driver's licenses registered anywhere in the world.
All of this goes back to a fundamental aspect of sampling problems. In any such problem, there must be a specified "population" of interest, for which we wish to make an inference, and there must be "sample data" that bears somehow on that inference. (Ideally, we would like the sample data to be a random sample from a sampling frame that is close to the population of interest.)
I think there is no way to know just by looking at the data.
In general, the population may be small or big and the sample may be small or big, hence in some situations the sample size might be quite close to the whole population. Imagine we would collect 90%, 95%, 99% and 100% of the population. I don't expect anything completely new happening with the results in case of the 100% (=population) data.
But maybe you know something about the population? If you know that the population consists of all customers of the company and you know how many customers they have per month you can maybe estimate how big the population is?
My question would be why you want to know that and why you don't know that? Usually one should know something about the data one is supposed to analyse. Keep in mind that inferencial statistics tries to draw conclusions about the population based on information that we know from the sample. This means if you have the population data there is no need for inferencial statistics (significance test, confidence intervals,...) and you can simply see the descriptive statistics. So such information about the data should be known by the analyst.
A sample is a just subset of the population. If the sample is representative (which it should be), the only main between sample and population is their size.
However, it should be noted that for any analysis in real life it's very important to know where the data comes from, and the process of collecting them needs to be well documented. Not even knowing whether the data is a sample looks like a rather bad red flag.