In a constitutional monarchy there is no problem. The prayer for the monarch is a prayer for the country. The monarch serve to represent the country.
In a republic, it is a bit trickier. The legal role played by the monarch (theoretically) in a place such as Britain, in America is played by the entire citizenry of the country (which includes us as well as we are citizens). The executive and legislature are mere servants to the people (analagous, using British legal theory, to the role of the parliament and ministers who in Britain are servants of the crown).
Many synagogues have a prayer for the government patterned after one used in Europe, and I suspect written by immigrants who hadn’t yet figured out that in American it is the people who rule the country, and the politicians are just employees hired to serve the people.