The verse actually quotes some of the wishful thinking of both Jews and Christians. Imam ibn Kathir in his tafsir also quoted other verses showing it:
The Hopes of the People of the Book
Allah made the confusion of the Jews and the Christians clear, since they claim that no one will enter Paradise, unless he is a Jew or a Christian. Similarly, Allah mentioned their claims in Surat Al-Ma'idah:
(We are the children of Allah and His loved ones) (5:18).
Allah refuted this false claim and informed them that they will be punished because of their sins. Previously we mentioned their claim that the Fire would not touch them for more than a few days, after which they would be put in Paradise. Allah rebuked this claim, and He said about this baseless claim,
(These are their own desires).
Abu Al-'Aliyah commented, "These are wishes that they wished Allah would answer, without basis." Similar was stated by Qatadah and Ar-Rabi' bin Anas. Allah then said,
(Say) meaning, "Say O Muhammad:"
("Produce your Burhan...")
meaning, "Your proof", as Abu Al-'Aliyah, Mujahid, As-Suddi and Ar-Rabi' bin Anas stated. Qatadah said that the Ayah means, "Bring the evidence that supports your statement,
(if you are truthful)
in your claim. "
Allah then said,
(Yes! But whoever submits his face (himself) to Allah (i.e. follows Allah's religion of Islamic Monotheism) and he is a Muhsin)
meaning, "Whoever performs deeds in sincerity, for Allah alone without partners." In a similar statement, Allah said,
(So if they dispute with you (Muhammad) say: "I have submitted myself to Allah (in Islam), and (so have) those who follow me.") (3:20) (Source: qtafsir)
So if these claims had any basis they would have a proof, but Allah () here and elsewhere challenges them to produce it if they had any!
On the contrary Allah () in (4:123) makes clear that reaching or even entering Jannah is not by wishful thinking, but only by good deeds and Allah's acceptance. (Also refer to Where in Qur'an does it say "Jannah will be in your accordance"?).
Sheikh Taher ibn 'Ashur in his tafsir called at-Tahrir wa at-Tanwir (see here in Arabic) التحرير والتنوير made clear that this statement shows the doubt of both parties Jews and Christians as the "or" or "أو" (in Arabic) in verse (2:111) is not one expressing that one must be either Jew or Christan to enter Jannah, but rather exclusive so each of them said you must be from among us to enter Jannah:
In the following I will translate from Arabic language, as these translations are of my own take them with the necessary care!
إلا من كان هودا أو نصارى ومقول القول مختلف باختلاف القائل فاليهود قالت لن يدخل الجنة إلا من كان هودا ، والنصارى قالت لن يدخل الجنة إلا من كان نصارى فجمع القرآن بين قوليهما على طريقة الإيجاز بجمع ما اشتركا فيه وهو نفي دخول الجنة عن المستثنى منه المحذوف لأجل تفريع الاستثناء .
(except one who is a Jew or a Christian) the stated words are different depending on the speaker, so Jews said: None will enter Jannah except one who is Jew, and the Christians said: None will enter Jannah except who is a Christian. The Qur'an here gathered both statements shortly in one statement that have in common the denial of entering Jannah for those who are excluded, but not named as they are many (different).
ثم جاء بعده تفريق ما اختص به كل فريق وهو قوله هودا أو نصارى فكلمة أو من كلام الحاكي في حكايته وليست من الكلام المحكي فأو هنا لتقسيم القولين ليرجع السامع كل قول إلى قائله والقرينة على أن أو ليست من مقولهم المحكي أنه لو كان من مقولهم لاقتضى أن كلا الفريقين لا ثقة له بالنجاة وأنه يعتقد إمكان نجاة مخالفه ، والمعلوم من حال أهل كل دين خلاف ذلك فإن كلا من اليهود والنصارى لا يشك في نجاة نفسه ولا يشك في ضلال مخالفه
Afterwards it quoted the differences between each party and this is the statement (Jew or a Christian) so the word "or" is from the speech of the transmitter of the story not from the made statements so that "or" here is to differentiate so that the listener would refer each statement to the specific speaker and the context that "or" was not among the words of their speech is based on the fact that if it was part of their speech then both parties don't have confidence in their salvation and they would believe the salvation of their counter-party, and the commonly known of each believer of a faith is the contrary of this. Either of Jews or Christians have no doubts in their salvation and no doubts in the deviation in his opponent.
The above statement shows that each religion would pretend the get salvation in the hereafter it is impossible that both can reach it, and therefore each party made the statement about its own salvation, not that of any other!
Note that in (2:78) the word "أَمَانِيَّ" (which I translated wishful thinking) was used before:
And among them are unlettered ones who do not know the Scripture except in wishful thinking, but they are only assuming.
Sheik ibn 'Ashur when interpreting this verse, interpreted it as follows: their knowledge of the Scripture was not profound, but their knowledge was mixed from what they heard and not from what they excelled (For details refer to the tafsir here in Arabic).
So basically the statements of both Jews and Christian have no strong basis either they are based on their hopes and wishful thinking that they might be the lonely ones that will be saved or based on the hear say that reached them from their incomplete knowledge of the scripture.