" I am the handmaid of the Lord." ( Luke1:38 )
Luke 1:38 could be translated the following way:
Because the Lord is my boss, he can demand whatever he wants and I have to do what he asks me.
I know that in Germany decades ago there were a lot of prayers in the Catholic church where the faithful were naming themselves as handmaids and servants of God.
This has the same meaning: The Lord is the boss of all people, so we have to do what the Lord wants us to do.
King Solomon and King David identified themselves both as servant although King and more so, both of them said they are the "son of thy handmaid"
Same meaning here: Both Kings say that the Lord is their boss ("I am your servant") and that he already was the boss of their parents ("son of your handmaid").
Your answer is purely personal opinion.
There is one aspect I forgot to mention, and this time it is definitely verifiable facts:
Luke wrote his gospel in Greek language while the Psalms are written in Hebrew language.
This means that it is hard to say if both texts use the same word ("handmaid") at all because when translating a text into another language the translator always has the choice between different words.
So it depends on the translators of the Bible if Luke 1 and Psalm 86 use the same word or if these texts use different words!
Comparing the Book of Wisdom to the Gospel is easier: Both books are written in Greek language.
Unfortunately I don't speak Greek. However, as far as I understand the words translated by Google Translator correctly, Mary uses the same word in Luke 1:38 that Solomon uses for himself (and not for his mother!) in Wisdom 9:5:
The same word is translated as "handmaid" in Luke 1:38 and as "servant" in Wisdom 9:5. Google Translate translates the word as "slave".
So arguing "it is the same word" will not work with translations but only with the original texts.
Does the word "boss" ever appear or been use in the scriptures to address God?
It seems to me that you think the Bible has originally been written in English language. However, this is not the case.
The Bible has been written in Hebrew and Greek languages and has been translated to English language.
In the original texts you will neither find the word "boss" nor the word "Lord" because these are English words.
It was the personal choice of the translators of your Bible translation to use the word "Lord" and not the word "boss" to address God.
Other translations use other words: Although English is not my native language, I know about one existing English translation of the Bible that does not use the word "Lord" at all!
And as far as I understand the Greek translations correctly, in the original text Solomon and Mary use the same word to describe themselves, but Solomon uses a different word describing his mother.
This means that your initial observation (Mary uses the same word for herself as Solomon uses for his mother) is only true for the English (and the German!) translation of the Bible, but it is not true for the original text written by Luke.
I doubt that you believe that a translation of the Bible written by some translator contains information that the original text written by Luke does not contain.
... failed to answer, why King David and King Solomon called their mother a handmaid, ...
They both identified themselves as "son of thy handmaid"
You didn't get the point:
Up to the 4th century, the Gospels and the book of Wisdom were only available in Greek language. But the original texts are looking like this (I hope I did not make a mistake because I don't speak Greek):
Book of Wisdom, chapter 9:
For I thy doulos (= male slave) and son of thine paidiske (= female servant) am a feeble person, ...
Gospel of Luke, chapter 1:
And Mary said, Behold the doule (= female slave) of the Lord; be it unto me according to thy word.
So the answer to your question is simple:
The two verses in the book of Wisdom and in the gospel of Luke DO NOT use the same word!
And of course the Psalm and the Gospel cannot use the same word because they are not written in different languages.
As far as I'm informed correctly, the first translation of the Bible that uses the same word in the different verses has been written about 300 years after Mary's death!
... or speaking in prophecy?
... a text that is written 300 years after some event is not a prophecy!