Now, to be 100% clear, anyone can baptize someone without being ordained. Whether it will be considered 'valid' depends on who you ask. The ceremony you'll see below meets all the requirements of the Roman Catholic Church, for example, in that the father uses water with the intention of doing what the church does in baptism, and using the correct words. That he mentioned the Universal Life Church is essentially irrelevant. However, if you ask the pastors of most nondenominational evangelical churches, or really any Baptist churches, they will tell you that this baptism is meaningless, because the child doesn't understand what is going on and the form is incorrect. Although water is used, the word for 'baptism' in the New Testament means 'immersion.' The early church only baptized consenting adults, and only by immersion, until sometime in the third century of the Common Era.
What matters here, however, is that the ceremony was meaningful for the family. As a Humanist celebrant I wouldn't christen a child, but I would gladly officiate a child welcoming if that's what someone wanted. These types of events can be valuable as ways to maintain a sense of family cohesion and tradition, while celebrating the addition of someone new to the family. In fact, in the event of an adoption a child welcoming ceremony could be very worthwhile as well.
In any event, do enjoy the video below, as it provides some perspective on how people make meaning and find ways to honor traditions without ceding ground to organized religions.