What does Sinclair Ferguson (a reformed theologian) mean by a 'new identity'?

We each have many identities (or aspects to our identity). In Christ, in the Gospel, we are given new identities, and our old ones are transformed. I'm sure you're familiar with all of these, just maybe not under the category of "identity", so I'll just give a brief overview.

One of the most important new identities we are given is that of the Adopted Child of God. Many verses speak of our adoption, including Ephesians 1:5 and Galatians 4:1-7.

One of the other new identities we are given is being Citizens of Heaven, from Philippians 3:20. We don't stop being citizens of the UK, or China, or Nigeria, etc. when we become Christians, but we are given a new heavenly citizenship that transforms how we think of ourselves as political citizens of our countries in this life.

Other identities change. We are still sinners, but we are redeemed and forgiven sinners. Some people's public identity is forever changed after one particular sin: the adulterer, the murderer, the sex offender, the cowardly traitor. These too can have their shame lifted as they realise their new identity in Christ, the forgiven sinner who is loved by God.

Other people feel a great significance in their sexual or gender identity. Sometimes these identities can cause us great distress, other times they are sources of great pride. Family is also another important aspect of many people's identities; some people have wonderful loving families, others deeply hurtful and dysfunctional families, and some may have no known family at all. In Christ we can experience peace with these aspects of our identity; they are not the entirety of who we are, and our identity as the people of God shapes and governs these other identities.

We find identity in our careers, our skills and passions, our accomplishments. Christ gives us new Gospel priorities and new Gospel purposes, to tell the world about him, to share God's overflowing love with the people of this world. When Christians give up profitable and prestigious careers to minister to drug addicts, people living in slums, people with dangerous contagious diseases, this shows that they have taken on a new identity, that of Christ's coworkers, who were willing to change the direction their life was heading in because they met Jesus.

I'm not aware of the specific lectures you are referring to, but I have listened to / read a fair amount of his work. I imagine Sinclair Ferguson would be thinking along these lines:

Therefore, if anyone is in Christ, the new creation has come: the old has gone, the new is here! (2 Corinthians 5:17)

I have been crucified with Christ and I no longer live, but Christ lives in me. The life I now live in the body, I live by faith in the Son of God, who loved me and gave himself for me. (Galatians 2:20)

The point is that coming to Christ is the most significant thing that can happen to someone - Jesus likens it to being "born again" (John 3:3).

This is how Paul describes the Christian life:

You were taught, with regard to your former way of life, to put off your old self, which is being corrupted by its deceitful desires; to be made new in the attitude of your minds; and to put on the new self, created to be like God in true righteousness and holiness. (Ephesians 4:22-24)

It's significant enough that we can talk about an "old self" and a "new self". Of course the word 'identity' isn't used - that's a more modern construct - but I think the concept is there.

I should add that it's not being given a new identity doesn't mean a totally new identity in the sense that it's alien to us: rather, it's the restoration of an identity which sin had distorted and defiled. In Christ we become the people who God designed us to be.

Without wishing to put words in Dr Ferguson's mouth, I think he is probably meaning something like that!