Yes, those two identities are conflicting with each other but this conflict is a positive Torah-inspired one.
The name “Israel” for the Jews means “God rules”, or “God shines”. Israel is a Biblical name. The patriarch Jacob was given the name Israel after he wrestled with the angel (Genesis 32:28 and 35:10). This biblical name comes from the Torah together with a set of moral/spiritual guidance for individuals, both Jewish and non-Jewish, for creating a “better world” for all humans with a special role for the Jews as the Chosen.
The name “State” is a political name. A State as a political organization links individuals to groups and organizations, and establishes balance and reconciliation between individuals and groups. Based on the founding document of a state, it provides security to individuals and safeguards the rights for individual humans of the state: it defends its citizens from all internal and external enemies who may be its own citizens or citizens of other states.
Thus the “Israel” identity calls for accepting and treating all humans with no discrimination while the “State” identity requires accepting and treating two categories of humans differently: “us” and “them”. The “us” are the humans who are citizens of the State of Israel while the “them” are those who are not the citizens and very often enemies who are trying to harm the State of Israel itself and its citizens.
In respect of these conflicting identities, the State of Israel is unique; there are no such conflicting identities in the names of other states. Here are a few examples. The State of USA, the State of Russian Federation, the State of China, the State of Germany, or the State of Saudi Arabia are defending and protecting their own national spiritual doctrines without trying to impose their doctrines on other nations. They may try to impose their doctrines on neighboring spiritually-close states but they don’t try to remake the entire world. (Of course there are numerous terrorist organizations who would like remaking the entire world but they are not states.) If they invade other states, it is done based on security considerations of a nation-state – not on spiritual considerations.
With the State of Israel everything is different. The state of Israel is a part of the Nation of Israel which is the Nation of the Chosen people – chosen by One God for assisting Him in spiritual remaking of the entire world from the pagan’s mind-set to the Torah-based mind-set. Such remaking is at the very foundation of Jewish mind-set even for those who proclaim they don’t believe in any Jewish Chosenness.
So what to do, how to manage those two conflicting identities?
The most important step in managing those conflicting identities is to recognize this conflict is a normal positive Torah-based conflict. The Torah itself consists of myriads of conflicting situations and the God’s truth is being searched for not through accepting “everybody has a right on his/her own opinion” but thought finding what is closer to the Torah guidance which was in the course of history transmitted to the Bible and then the Declaration of Independence.
However, many Jews and non-Jews don’t believe those conflicting identities are a normal state of human Torah-based development and an example of that could be seen from a post of Rabbi Michael Melchior “Jewish State bill is precisely what makes the prophets weep” at https://blogs.timesofisrael.com/jewish-state-bill-is-precisely-what-makes-the-prophets-weep/#ixzz3K6oBHbYd where he writes:
“Of all the mitzvot (religious laws) in the Torah, Hillel the Elder chose to tell the convert, while he was standing on one foot, that the entire Torah in a nutshell comes to: “That which is hateful to you, do not do to your fellow.” Haven’t we learned anything from that? The attempt to oust the other — the Arab minority — from Israeli democracy is not a question of nice clothes or ugly clothes. It’s just not clothes at all. And it’s not only the mitzvot in the Torah that teach us that this motion is anti-Jewish. Jewish history itself, in which the Jews lived as a minority in lands not their own, shows the way, beginning with: “Love the stranger: for you were strangers in the land of Egypt.” And if this was said regarding the Israelites in Egypt, then it is all the more true regarding Arabs in Israel who are a part of this place. Here is their home and here is their future — along with us. That is why I maintain that the “Jewish State bill” is a ringing slap to any Jewish basis of the state.”
I am not a rabbi but I have been studying the Torah and Judaism for many-many years, and I find in the Torah something completely different in regard to the mitzvoth “That which is hateful to you, do not do to your fellow” and “Love the stranger: for you were strangers in the land of Egypt.”.
The definitions of “your fellow” and “the stranger” don’t include your true enemies since your true enemies are enemies of the Torah itself. Your true enemies are true enemies of the Torah since they are trying to create/remake the world along the lines of anti-Torah world design. They are the Evil and you are obliged to fight the Evil. Hitler and his Nazi-nics, Stalin and his Communist-nics, Nasrallah and his Hezbollah-nics, Khaled Meshaal and his Hamas-nics are not “your fellow” or “the stranger” – they are the Evil in a Torah-based definition.
Your enemies are not “your fellow” – “your fellow” is one who is trying to create a Torah-based better world even without recognizing it is a Torah-based one. “Your fellow” may be a Jew, a Christian, an Arab, a Hindu, a Buddhist, or even a Muslim if his/her individual faith has a Torah-based definition of “your fellow”. However, somebody who is trying to annihilate you physically or spiritually is not a Torah-based “your fellow”.
Your enemies are not “the stranger” – “the stranger” is one whom you unexpectedly meet and rightly assume he/she is supposed to be a good human being created in the image and likeness of God. “The stranger” may be any human being of any tribe or beliefs. However, somebody who is openly trying to eliminate you physically or spiritually is not a Torah-based “the stranger”.
Most of the Muslim states surrounding the State of Israel proclaimed their goal as eliminating the Jewish presence in the Middle East. They may agree with a Sharia-based State of Israel but never with the Torah-based State of Israel. Therefore proclaiming in a state law the Jewish, Torah-based nature of the State of Israel is extremely important for spiritual survival of the State of Israel and the entire Jewish nation.
Thus the two conflicting identities of the State of Israel – the identity of the State and the identity of Israel – will be conflicting (as it is supposed to be in accordance with the Torah) until the coming of Messiah … in his religious or secular definition.