We are providing an app which manages your finances — like a housekeeping book. Our space is limited; therefore, we want to use the slogan "Housekeeping App". Would an English speaker know what is meant by that?
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To expand on my comment, housekeeping to me has the wrong connotation. Perhaps the word you're looking for is bookkeeping? (Although that arguably has some negative connotations with regards to gambling, I reckon.)
Edit to add: Another "word" that might work for you is HouseBook. I've never heard the term before, but it might allow you to allude to the specific nature of this App, while also including a nod to bookkeeping. As Steve mentions, in the product description you should refer to personal finance.
If you mean that, just by the slogan "housekeeping app", an English speaker would be likely to think you are talking about an app that manages finances, then the answer is definitely no.
The primary meaning of housekeeping is cleaning and repairing the house. Keeping track of personal finances is not generally considered part of housekeeping at all.
If anything, I would consider the management of finances to be "finance tracking".
A general term for software which allows you to manage your own finances would be personal finance software.
There are examples of references to such software on other Stack Exchange sites, for example:
As several others have mentioned, "housekeeping" is not the right word here, as it doesn't convey any association with money.
However, an app called Accountant might convey the right idea.