Wales, Scotland, and England have their own teams (Northern Ireland contributes to the Irish team). The common explanation I’ve seen is that the nations have deep, distinct histories and cultures, but that doesn’t explain why, for example, India doesn’t get to send a team from each state despite India’s ethnic groups being even more diverse and distinct than those of the UK.
So what is the actual reason for Britain into three parts for the purpose of international sport but not other countries?
Because the UK is where the sport originated and they already had separate national teams before anyone knew soccer would one day be an internationally successful sport and founded organizations like FIFA to organize it internationally. The idea of an international soccer match started with England playing against Scotland. So they got to keep that as an exception.