Christ became the man Jesus, a mortal, subject to death the same as we are. He really did die. He was resurrected and became immortal and now lives. That's why the bible says he alone has immortality. Nobody else has been resurrected to immortality:
1Co 15:20 But now is Christ risen from the dead, and become the firstfruits of them that slept.
1Co 15:21 For since by man came death, by man came also the resurrection of the dead.
Seems you're simply not distinguishing between the death of the body and the death of the soul.
Souls die:
Jam 5:20 Let him know, that he which converteth the sinner from the error of his way shall save a soul from death, and shall hide a multitude of sins.
Mat 10:28 And fear not them which kill the body, but are not able to kill the soul: but rather fear him which is able to destroy both soul and body in hell.
Eze 18:4 Behold, all souls are mine; as the soul of the father, so also the soul of the son is mine: the soul that sinneth, it shall die.
Eze 18:20 The soul that sinneth, it shall die. The son shall not bear the iniquity of the father, neither shall the father bear the iniquity of the son: the righteousness of the righteous shall be upon him, and the wickedness of the wicked shall be upon him.
But of course, the Scriptures state that the wages of sin is death; death is the consequence of sin. Therefore, to say that Christ was subject to death just as we sinners are is to posit that Christ was a sinner, which clearly is not the case.
Christ is both true God and true man, was sinless, and would not have died unless it was by violent action enacted upon Him by others.
Eternal death, yes. Do you believe that this death of soul equates to a ceasing-to-be?