We later find out that this nurse was called Devorah as we learn about her death in Bereishis 35:38. The pasuk over there expressly calls her the nurse of Rivka.
It would make sense to assume that she was sent at the time to care for Rivka, owing to the fact that Rivka got married at such a young age. Indeed the Targum Yonasan translates the word 'מֵֽנִקְתָּ֑הּ' - 'nurse' as פַּדְגְוָותָהּ stemming from the word 'pedagogy', meaning to say that her role was that of an educator. Moreover, this relationship was one of care and not of servitude like that of a 'shifcha' - maidservant like you ask. The Shadal writes on the pasuk 24:59:
מנקתה: בימים הקדמונים (ראב"ע), ונראה שהיו המיניקות נשארות בבתי העשירים אצל יונקותיהן (גם אחר שגדלו) באהבה ורעות, ולא שלחו המינקת לשמשה כשפחה, כי כבר שלחו עמה גם נערותיה (פסוק ס"א).
Her nurse: In the early days, it appears that nurses remained in the houses of the wealthy by their infants (even after they grew up) with love and friendship. And they didn't send the nurses to serve like maidservants because they had already sent with her her lads.
So it would seem from this explanation that she definitely was not serving in a 'shifcha' capacity but rather maintained an ongoing relationship with Rivka by virtue of being hired by a wealthy home.
This loyalty continued to the end of her life. If we return to the pasuk that I started with in which the Torah quotes her death. Rashi tells us:
ותמת דבורה. מָה עִנְיַן דְּבוֹרָה בְּבֵית יַעֲקֹב? אֶלָּא לְפִי שֶׁאָמְרָה רִבְקָה לְיַעֲקֹב וְשָׁלַחְתִּי וּלְקַחְתִּיךָ מִשָּׁם, שָׁלְחָה דְבוֹרָה אֶצְלוֹ לְפַדַּן אֲרָם לָצֵאת מִשָּׁם, וּמֵתָה בַדֶּרֶךְ; מִדִּבְרֵי רַבִּי מֹשֶׁה הַדַּרְשָׁן לְמַדְתִּיהָ:
AND DEBORAH DIED — How came Deborah to be in Jacob’s house? But the explanation is: because Rebékah had promised Jacob (Gen. 27:45) “then I will send and fetch thee from thence”, she sent Deborah to him to Padan-aram to tell him to leave that place, and she died on the return journey I learnt this from a comment of R. Moses Ha-darshan. (sefaria translation)
So Rivka after sending Yaakov away from home to flee an enraged Eisav, is now fulfilling her promise to notify Yaakov that the time was now right for him to return home (Also refer to the Chizkuni). Hence, the ever dutiful Devorah right to the end, is in the service of Rivka as she and the household of Yaakov make their return to her mistress Rivka.