I agree with your comment that any sin kills faith and leads to death.
We do not believe all sins are equal though. For example a mortal sin for us requires one to know that it is a sin and then to actively pursue this sin. We believe this is a rejection of God and as it is an act that repudiates God it separates us from God.
The verse I sited is one of the verses we use to support the concept that all sins are not equal.
Do Lutherans believe in Mortal Sin in the same sense?
Yes I would say we do. We don’t categorize them into lists of sins which are mortal and venial. Here is a description from the Lutheran Church - Missouri Synod website (which is in agreement with the Synod I belong to, as well):
"In the case of believers, those sins are called mortal that force the Holy Spirit to depart from one’s heart and life–sins that destroy faith. Venial sins are sins which, though they in themselves merit eternal death, are daily forgiven to the believer. Venial sins are perhaps better called “sins of weakness.” They do not drive away the Holy Spirit, and do not extinguish faith.
We have a perfect example in the Bible of an individual who, as a believer, committed mortal sin: King David. When he took Uriah’s wife and committed adultery with her, and then schemed to have Uriah killed, he was actually in a state where he had fallen from faith and had driven out the Holy Spirit. These were no mere sins of weakness on his part, but sins knowingly committed, with great forethought and planning. The only solution to this grave situation was the preaching of God’s Law, which worked mightily in his life to move him to repentance and restoration to faith and God’s saving grace.
In unbelievers, all sin is without any distinction; they are always dominant, even though individual sins may not turn into actual actions. All unbelievers are dead in trespasses and sin, and Satan is the ruling power in them (Eph. 2:1-3; Col . 1:13; Acts 26:18).
The state of grace in which sin no longer rules over us is found only in believers (Romans 6:14). When in the lives of believers the struggle of the spirit against the flesh has ceased and the sins have again become dominant, then these believers have fallen from faith. Hence the admonition, “Let not sin therefore reign in your mortal body” (Rom. 6:12).
It is usually the case that faithful Christians are the ones most troubled by their sin, as well it should be, but they begin to wonder, “Have I fallen so far that God’s grace can not reach me?” In that case, of course, the answer is, “No.”
And it is also the case that those caught up in a sinful lifestyle that is driving them further and further away from the kingdom may not recognize their sin. To those people, God’s word of Law must be clearly proclaimed so that, like David, they are turned from their sin and back to the mercy of God, given us through our Lord Jesus Christ, whose blood cleanses us from all sin."