If the mother of my child enrolls our son at a daycare and tells them to use a different name other then his real name, can she or can that company get in trouble for using a fake name for a child?

Depends on the reason for doing so. Also depends on the nature of the day care.

In America (ANY question on law needs to include your general locale as laws vary by area. ) a person can use any name they want as long as there is no intent to defraud. So, unless the day care provider is required by local law to see identification (which may be a requirement) they may not be responsible.

If the real ID of the child is provided but the mother requests that the child be known as Jimmy instead of Sebastian, that is not illegal. If, as I suspect, the issue is that the mother wants the child referred to as John Doe instead of John Smith (Smith being your last name), that is also not illegal, though it could depend on the type of custody you have.

So, the likelihood is that no, there is no trouble here. But if you want to explain the circumstances in greater detail we may have a different opinion.