The context is also in Hebrews 3, where people perished due to their unbelief and disobedience 40 years in the wilderness. 40 years in the wilderness, they were offered a literal 7th day rest, not an eternal rest. Therefore the Sabbath-rest is in reference to a literal 7th day Sabbath.
So do you believe in the same Sabbath rest that those who, because of unbelief and disobedience, perished in the wilderness for? The same Sabbath they perished for was a literal day; they weren't offered a Sabbath rest in Christ. So "Jesus is my Sabbath" is an invalid point for Hebrews 3-4.
Here are the two verses from Chapters 3 and 4.
Hebrews 3:18-19
And to whom did He swear that they would not enter His rest, but to those who were disobedient?
And so we see that they were not able to enter because of unbelief.
Hebrews 4:1-3
Therefore, we must fear if, while a promise remains of entering His rest, any one of you may seem to have come short of it.
For indeed we have had good news preached to us, just as they also did; but the word they heard did not benefit them, because they were not united with those who listened with faith. For we who have believed enter that rest, just as He has said.
It is very easy to see when you put the verses together.
The subject is the good news, Jesus Christ.
Belief or unbelief in God's promise, Jesus Christ.
The rest is eternal life that God promised in Jesus Christ.
The sabbath rest is not the real issue being discussed here. The fulfillment of the sabbath rest was always the promise. As the fulfillment of all the law and the prophets as only Jesus could fulfill.
The subject of Hebrews is Jesus Christ, the Good News.