Is it considered a lie for an employee to register as married and obtain housing designated for married couples to avoid social embarrassment, knowing that he could obtain housing without mentioning bringing children? And is his acquisition of housing in this case unlawful?

Lying is informing contrary to reality. Registering yourself as married when you are not is considered a lie. If this lie leads to obtaining privileges you do not deserve (such as a salary increase or housing for married individuals), then this increases the sin and makes it unlawful to take. You must repent to Allah for the lie you uttered.