Is inequality a micro or macro economics issue?

Inequality and public policy has actually its own field (Public Economics) so it does not narrowly fit in either of those categories, but given that most research and work on inequality is of microeconomic character (see Stiglitz Economics of the Public Sector 3rd ed) I think it is fair to put inequality under microeconomics if you have to make such binary choice as it fits there better.

First let us get the definitions of micro and macroeconomics straight. Following Mankiw Principles of Economics pp 27:

Microeconomics is the study of how households and firms make decisions and how they interact in specific markets.

Macroeconomics is the study of economy-wide phenomena.

Neither of them is top-down or bottom-up, in fact most macroeconomics models are build upon micro-foundations (in fact even you do research in pure macro you need to learn a lot of microeconomics), so by your descriptions in your question they would both be 'bottom-up' - although I do not think it is sensible to talk about these fields as top-down or bottom-up. The distinction is based rather on the scale at which you are looking.

When it comes to inequality, often the appropriate level of analysis is microeconomic analysis. For example, do you worry about income inequality? Well income inequality arises from the fact that people are heterogeneous in their ability. Hence you need to study how this affects rewards they get for their labor supply on a labor market. Such analysis does not require tracking of inflation and other macro factors that would likely just 'muddy the waters'. To study that you want some micro models not macro models where everyone is aggregated to an 'representative agent' (some sort of average human). Similar arguments would apply to most topics to be considered in the realm of income inequality.

Macroeconomics, also touches bit on inequality as there can be interplays between inequality and economic growth (and growth is definitely macro topic). Or unemployment benefits or redistribution from people who have high marginal propensity to save (usually rich people) to ones with low propensity to save (usually poor) can be considered macroeconomic stabilizers. But macroeconomics does not really concern itself with inequality per se. If it would empirically turn out that poor people have higher propensity to save than rich people from pure macro perspective there should be redistribution from poor to the rich.

This is not to say that there is absolutely no research into inequality where inequality would be in the main focus and the research is just using macro models/tools. But this strand of literature is miniscule compared to microeconomic literature devoted to this topic.

For example, consider this. In Romer Macroeconomics, which is the standard widely used graduate macro textbook the word income tax is printed only 8 times. In the Varian Microeconomic Analysis you have separate sub chapter on optimal taxation, separate chapter for welfare analysis, separate chapter for public goods etc.

This is not because macroeconomists would want to ignore inequality but it is simply not considered to be part of the subject matter. This primarily comes to the fact that macroeconomists often have to aggregate when they try to describe whole economy. In order to aggregate you need to introduce average households, average individuals, average firms etc. Once you aggregate whole population to set of average individuals then income/wealth inequality amongst those averages is zero.

There are macro models that also look heterogeneous agents but even there you will usually have just some small limited number of options. For example, having two households that are different and each is supposed to be an avatar of their own category. It is possible to still introduce more heterogeneity into macro models but the more heterogeneity you include the complexity of the model raises disproportionately. This is less problematic in microeconomic models and thus they are preferred.

Lastly, it is possible that in the future this will change and people studying inequality and policy economics in general will start focusing more on macro models and treating inequality as a macro phenomenon. From my own experience work on inequality from macro side is nowadays more common then when I was undergraduate. Nonetheless, as of now I don't think anyone could reasonably disagree with statement that most inequality work focuses solely on micro (for example, almost whole literature on optimal income taxes and transfers is microeconomics) or uses micro models. Hence, if you have only two options I think it makes more sense to put it under microeconomics and I think this was why the author did that.