You are correct. The following passage teaches us that God counts our faith as righteousness, but it does not imply we are credited with the righteousness of Christ...
2 Corinthians 5:21 says, "For He made Him who knew no sin to be sin for us, that we might become the righteousness of God in Him." It speaks of "becomming" the righteousness of God in Him. It says nothing about being credited with His righteousness.
Abraham Justified by Faith
4 What then shall we say that Abraham our father has found according to the flesh? 2 For if Abraham was justified by works, he has something to boast about, but not before God. 3 For what does the Scripture say? “Abraham believed God, and it was accounted to him for righteousness.” 4 Now to him who works, the wages are not counted as grace but as debt.
David Celebrates the Same Truth
5 But to him who does not work but believes on Him who justifies the ungodly, his faith is accounted for righteousness, 6 just as David also describes the blessedness of the man to whom God imputes righteousness apart from works:
7 “Blessed are those whose lawless deeds are forgiven,
And whose sins are covered;
8 Blessed is the man to whom the LORD shall not impute sin.”
The New King James Version (Ro 4:1–8). (1982). Thomas Nelson.
Note: The bold+underline is where the underlying original language uses the word, "λογίζομαι", which is translated here "counted", "accounted", "impute" and "imputes".
2 Corinthians 5:21 says, "For He made Him who knew no sin to be sin for us, that we might become the righteousness of God in Him." It speaks of "becomming" the righteousness of God in Him. It says nothing about being credited with His righteousness.
God's righteousness demands perfection. Has there ever been on person who has claim to that perfection? Abraham also disobeyed God so his belief, while righteous in it's acceptance of God, cannot possibly achieve the total perfection of God. I am saying Abraham was not perfect which would be the requirement before God for acceptance before Him.
A person can be righteous one moment and sin the next. That, is not righteous nor acceptable for God's requirement for the penalty of sin, which is death. The only way to obtain this righteousness is through Jesus, His death on our behalf. That, is what is meant by 'we have the righteousness of Christ' as scripture declares.
We have all become like one who is unclean, and all our righteous deeds are like a polluted garment. We all fade like a leaf, and our iniquities, like the wind, take us away. Isaiah 64:6
Abraham was long dead when Isaiah declared the above. No mention of any type of righteousness a person may make before God. Rather the declaration that even what we say or consider as righteous, is vile to God.
Since, therefore, we have now been justified by his blood, much more shall we be saved by him from the wrath of God. Romans 5:9
Justification means we are not considered as sinful beings before God as our Father in heaven now sees us as sinless because of the perfect sacrifice of Jesus. People sometimes have a problem with that concept and will declare 'oh but I still sin!' Well, that is why Jesus came to earth to die for us. His sacrifice is what is acceptable to God on our behalf. We have forgiveness in Jesus.