Wa Alaikum As Salaam,
It is permissible to engage in sports for the purpose of relaxation, satisfaction and enjoyment, as long as one does not violate the teachings of Islam by exposing his awrah, (concealable parts of the body), missing/delaying his duties of Salah or fasting etc., and is not any of those sports which have been prohibited in the Shariah.
With respect to the hadith you have quoted, this is meant to encourage the believers in such sports /training. Archery and horse-training helps one in preparing himself to defend Allah’s religion when the time arrives, and playing with one’s wife builds the bond/relationship between the husband and the wife and brings joy and happiness in the marriage. Other things have also been mentioned in other traditions like walking, swimming, spear throwing, horse-racing and others. All these sporting games bring a lot of benefits to a person and so, they are encouraged. However, this does not mean that other types of sports are prohibited or disliked. These will be permissible as long as they do not comprise of anything which violates the laws of the Shariah.
And Allah Knows best
Mufti Waseem Khan