Hi Weeys,
Employee engagement activities are usually social in nature. It involves all those activities which allow the employees to interact with each other, play some games, or simply pass time discussing non-office topics. This helps them fulfill their social needs and is a great motivator for them.
In our office, we celebrate all the birthdays of our employees on the 15th of each month. We order food and cake for everyone, and there is a party on that day. We also celebrate major festivals as ethnic day or with some special dress codes.
I guess it can be said that all the social activities done in and for the office are largely covered under employee engagement. However, social activities have a much larger spectrum than employee engagement.