Were Adams and Jesus‘ Spirit part of similar events?

I can contribute from my comparative studies on Quran and Bible.

Referring to Jesus ﷺ:

Quran 21:91

وَالَّتِي أَحْصَنَتْ فَرْجَهَا فَنَفَخْنَا فِيهَا مِن رُّوحِنَا وَجَعَلْنَاهَا وَابْنَهَا آيَةً لِّلْعَالَمِينَ

And she who guarded her virginity. We breathed into her of Our spirit, and made her and her son a sign to the world. (Itani)

Also found in Quran 9:17

فَاتَّخَذَتْ مِن دُونِهِمْ حِجَابًا فَأَرْسَلْنَا إِلَيْهَا رُوحَنَا فَتَمَثَّلَ لَهَا بَشَرًا سَوِيًّا

She screened herself away from them, and We sent to her Our spirit, and He appeared to her as an immaculate human. (Itani)

The passages confirm the message related in the Bible,

Matthew 1:18

This is how the birth of Jesus Christ came about: His mother Mary was pledged in marriage to Joseph, but before they came together, she was found to be with Child through the Holy Spirit.

Modern Arabic translation (Online Bible):

اما ولادة يسوع المسيح فكانت هكذا.لما كانت مريم امه مخطوبة ليوسف قبل ان يجتمعا وجدت حبلى من الروح القدس.

The Holy Spirit in Christian understanding is the Spirit of God. It is this Spirit that helps the disciples to talk and spread out the words and deeds of Jesus ﷺ after ascension:

Acts 1:8

But you shall receive power when the Holy Spirit has come upon you; and you shall be my witnesses in Jerusalem and in all Judea and Sama'ria and to the end of the earth."

Modern Arabic:

لكنكم ستنالون قوة متى حل الروح القدس عليكم وتكونون لي شهودا في اورشليم وفي كل اليهودية والسامرة والى اقصى الارض

Now we read in Quran 16:102

قُلْ نَزَّلَهُ رُوحُ الْقُدُسِ مِن رَّبِّكَ بِالْحَقِّ لِيُثَبِّتَ الَّذِينَ آمَنُوا وَهُدًى وَبُشْرَىٰ لِلْمُسْلِمِينَ

Say, “The Holy Spirit has brought it down from your Lord, truthfully, in order to stabilize those who believe, and as guidance and good news for those who submit.” (Itani)

It is the same Holy Spirit that came over the Prophet Muhammad ﷺ, there perceived as the Angel Gabriel. Hence, to translate the Holy Spirit with the Angel Gabriel is legitimate (although I personally prefer translations that abstain from unnecessary interpretations).

Now to the second passage:

Quran 15:29

فَإِذَا سَوَّيْتُهُ وَنَفَخْتُ فِيهِ مِن رُّوحِي فَقَعُوا لَهُ سَاجِدِينَ

“When I have formed him, and breathed into him of My spirit, fall down prostrating before him.” (Itani)

It refers to Genesis 2:7

וַיִּיצֶר יְהוָה אֱלֹהִים אֶת־הָאָדָם, עָפָר מִן־הָאֲדָמָה, וַיִּפַּח בְּאַפָּיו, נִשְׁמַת חַיִּים; וַיְהִי הָאָדָם, לְנֶפֶשׁ חַיָּה׃ Then the LORD God formed man of dust from the ground, and breathed into his nostrils the Spirit of Life; and man became a living being.

We also find the translation

Then the LORD God formed man of dust from the ground, and breathed into his nostrils the breath of life; and man became a living being.

Modern Arabic translation (Online Bible) translates נִשְׁמַת = نَسَمَة . However, نَسَمَة is never used in the Quran. We always find the word رُوح. This word also exists in Hebrew, as in Gen 1:2

וְהָאָרֶץ, הָיְתָה תֹהוּ וָבֹהוּ, וְחֹשֶׁךְ, עַל־פְּנֵי תְהוֹם; וְרוּחַ אֱלֹהִים, מְרַחֶפֶת עַל־פְּנֵי הַמָּיִם׃

The earth was without form and void, and darkness was upon the face of the deep; and the Spirit of God was moving over the face of the waters.

The words have similar meanings; נִשְׁמַת is translated as breath or Spirit, רוּחַ as wind or Spirit. Arabic and Hebrew are the same word with the same letters RWH and the same meanings.

רוּחַ = رُوح

In Greek, words with the same stem are used for the breath/Spirit of Life, the Spirit of God and the Holy Spirit:

Genesis 1:2 ἡ δὲ γῆ ἦν ἀόρατος καὶ ἀκατασκεύαστος καὶ σκότος ἐπάνω τῆς ἀβύσσου καὶ **πνεῦμα θεοῦ ἐπεφέρετο ἐπάνω τοῦ ὕδατος

Genesis 2:7 καὶ ἔπλασεν ὁ θεὸς τὸν ἄνθρωπον χοῦν ἀπὸ τῆς γῆς καὶ ἐνεφύσησεν εἰς τὸ πρόσωπον αὐτοῦ πνοὴν ζωῆς καὶ ἐγένετο ὁ ἄνθρωπος εἰς ψυχὴν ζῶσαν

Matthew 1:18 τοῦ δὲ ἰησοῦ χριστοῦ ἡ γένεσις οὕτως ἦν. μνηστευθείσης τῆς μητρὸς αὐτοῦ μαρίας τῶ ἰωσήφ, πρὶν ἢ συνελθεῖν αὐτοὺς εὑρέθη ἐν γαστρὶ ἔχουσα ἐκ πνεύματος ἁγίου.

Acts 1:8

Greek: ἀλλὰ λήμψεσθε δύναμιν ἐπελθόντος τοῦ ἁγίου πνεύματος ἐφ᾽ ὑμᾶς, καὶ ἔσεσθέ μου μάρτυρες ἔν τε ἰερουσαλὴμ καὶ [ἐν] πάσῃ τῇ ἰουδαίᾳ καὶ σαμαρείᾳ καὶ ἕως ἐσχάτου τῆς γῆς.

Modern Arabic:

لكنكم ستنالون قوة متى حل الروح القدس عليكم وتكونون لي شهودا في اورشليم وفي كل اليهودية والسامرة والى اقصى الارض

Aramaic: ܐܠܐ ܟܕ ܬܐܬܐ ܪܘܚܐ ܕܩܘܕܫܐ ܥܠܝܟܘܢ ܬܩܒܠܘܢ ܚܝܠܐ ܘܬܗܘܘܢ ܠܝ ܣܗܕܐ ܒܐܘܪܫܠܡ ܘܒܟܠܗ ܝܗܘܕ ܘܐܦ ܒܝܬ ܫܡܪܝܐ ܘܥܕܡܐ ܠܣܘܦܝܗ ܕܐܪܥܐ

So we find essentially

ܪܘܚܐ=روخا=רוּחַ = رُوح =Spirit = πνεύμα

as well as the very similar group

נִשְׁמַת = رُوح =Spirit = πνοὴν

In both, and other cases, it is the Spirit of God given to a Human. You are right that the Quran confirms the Bible saying that Jesus and Adam were created by teh Holy Spirit of God, contradicting the false interpretation of the Nicean Credo, stating that Jesus was genitus, non factus.

With our modern knowledge that in fact the cration of Adam is an allegory, saying that out souls come from God, the meaning of this act of giving both is equivalent for Adam and Jesus.