Seeking an open dataset listing the number of banking institutions that exist, and the country/countries they exist in?
Data is collected by the International Monetary Fund, Financial Access Survey. Data has a friendly license.
On the right you will see a big DOWNLOAD button (CSV, XML or XLS)
If, for some reason, you don't want the data per capita, but instead total bank branches, then you can use this data set to get the population of each country.
If you want the number of bank "companies" per countries, then you can harvest this data from Wikipedia
On the left menu-bar of every Wikipedia page is a link to "Wikidata Item". You can then use a Wikidata query tool to programmatically collect the data. See other questsions or do an internet search about Wikidata or SPARQL query language.
Or, use this website Banks by country for EU banks (including offshore territories).