1. Question Restatement:
Strong's equates the name "Jesus" with the Hebrew name "Joshua or Johoshua." On what basis does he do this?
Quick Answer: This was a standard convention, as seen in the Greek Septuagint Translation of Hebrew Scripture.
2. The Transliteration and Meaning of Jesus in Greek:
The name Jesus, ("Ἰησοῦς", Strong's Concordance), in Greek, doesn't hold any significance - and doesn't appear as a common Greek name, or word, in Greek Literature, (See Perseus Greek Word Search).
However: "Jesus / Ιησος" in the Greek is actually a very good transliteration of Jesus' name from the Hebrew/Aramaic "isho' / ܝܘܣܐ", (Luke 3:29).
Note 1: In ancient Hebrew, and Aramaic, Jesus' name ends with a "guttural stop" - not an "A".
Note 2: Adding the final sigma, "ς" is a Greco-Roman convention to indicate the Nominative case - and not part of the transliteration. (In English, "Jesu + s").
3. Basis to Transliterate Joshua and Jehoshua as "Jesus":
Jesus is actually a Greek transliteration of an Aramaic/Hebrew name "Joshua" which is actually a the transliteration of "יְהוֹשׁ֫וּעַ | Yehoshua".
It is often claimed the "Jesus" is a transliteration of "Yeshua | יֵשׁ֨וּעַ" - but even that name is considered an abbreviated/transliterated form of: "יְהוֹשׁ֫וּעַ | Yehoshua".
3.1. For Reference: "Jesus" in Greek:
NASB, Matthew 1:1 - The record of the genealogy of Jesus the Messiah, the son of David, the son of Abraham:
Byz, Matthew 1:1 - βιβλος γενεσεως ιησου χριστου υιου δαυιδ υιου αβρααμ
3.2. Yeshua | ישוע is Derived from "Yehoshua", Which is Derived from "Hosea" :
Scripture explicitly states that "Yehoshua" is actually derived from the name, "Hosea | הוֹשֵׁ֫עַ", (Numbers 13:16) - which is derived from "יָשַׁע | he saves".
NASB, Numbers 13:16 - but Moses called Hoshea the son of Nun, Joshua.
Septuagint, Numbers 13:16 - καὶ ἐπωνόμασεν Μωυσῆς τὸν Αυση υἱὸν Ναυη Ἰησοῦν
Hebrew, Numbers 13:16 - וַיִּקְרָ֥א מֹשֶׁ֛ה לְהֹושֵׁ֥עַ בִּן־ נ֖וּן יְהֹושֻֽׁעַ׃
3.3. "Yeshua" is an Abbreviated / Later Aramaic form of "Yehoshua" :
NKJV, Nehemiah 8:17, Hebrew - ... for since the days of "Joshua | יֵשׁ֨וּעַ", the son of Nun ... ("Ἰησοῦ", LXX, Esdras-Nehemiah 8:17).
"Yehoshua" was most probably shortened to "Yeshua", (removing "yah*") - during the Babylonian exile - plausibly in an effort to avoid misusing the name of God - a tradition that is observed to this day.
In Scripture, "Yeshua" does not appear as an Israeli name - until the Babylonian exile, (~597 b.c; see Occurrences of Yeshua as a name.)
3.4. More Examples of "Yeshua" and "Yehoshua" Transliterated as "Jesus":
NASB, Deuteronomy 3:21 - I commanded Joshua at that time, saying,
Hebrew, Deuteronomy 3:21 - וְאֶת־ יְהֹושׁ֣וּעַ צִוֵּ֔יתִי בָּעֵ֥ת הַהִ֖וא לֵאמֹ֑ר
Septuagint, Deuteronomy 3:21 - καὶ τῷ Ἰησοῖ ἐνετειλάμην
Note: 2 Chronicles likely written after 540 B.C., Wikipedia.
WEB, 2 Chronicles 31:15 - Under him were Eden, and Miniamin, and Jeshua
Septuagint, 2 Chronicles 31:15 - διὰ χειρὸς Οδομ καὶ Βενιαμιν καὶ Ἰησοῦς
Hebrew, 2 Chronicles 31:15 - וְעַל־ יָדֹ֡ו עֵ֣דֶן וּ֠מִנְיָמִן וְיֵשׁ֨וּעַ וּֽשְׁמַֽעְיָ֜הוּ
3.5. Yeshua is NOT Directly Derived from the Hebrew, "יָשַׁע":
In Hebrew Scripture, "He save(s, ed, etc)" ... does not appear to be ever rendered as "Yeshua", (with the "vav | ו").
This appears to be an Aramaic convention.1
Different forms are used, from the root, "יָשַׁע", as: "מ֝וֹשִׁ֗יעַ" (Psalms 7:10), "הוֹשִׁ֥יעַ" (Psalms 18:41), etc. (See Other Occurrences of "יָשַׁע.)
And so it is invalid to suggest that "Yeshua" strictly means, "He saves" - as it would likely be rendered without the "וּ" - : "וַיֹּ֣שַׁע", (Job 5:15); "מ֝וֹשִׁ֗יעַ", (Psalms 7:10); etc.
"לִֽישׁוּעָ֑ה | I will come to help", (2 Samuel 10:11), may be the closest.