This is a complete misunderstanding of some sects of Krishna devotions. Those sects which use the term "servant", meaning those which do not practice Karma Yoga with Bhakti Yoga as a path to Jnana Yoga, and have a different concept of Seva that zeros in on Prema Bhakti, they are not speaking of slavery but a type of relationship based on Love or Prema. I guess love can be construed as slavery, but slavery is involuntary, love is different indeed.
These sects have many technical terms about the intrinsic natures of this love, including servant, but they also use the term "I am the servant of the servant". What does this mean? Let me exp,ain.
I do not want to go the nuances of such technical terms of the types of relationships with Krishna, let the Gaudiyas do that since I am a Saiva. But it is not slavery. Slavery state does exist in Hinduism, for example in the sects of Manasa or Snake Goddess where She is VERY possessive of Her devotees and refuses to share their attention such that She will project a mystic on Her devotees where slavery to the Gauri is indeed expected, but this does not exist in these Krishna sects. Avoiding the nuances, since I am a Saiva, let me just cut to the chase and make it simple.
The servant of the servant means within such Krishna sects of Prema Bhakti, ASSOCIATION with Krishna devotees is VERY important. So such a devotee sees every other devotee as superior, thus they think to serve the devotees is "ecstasy", a statement of humility to such devotees by another devotee in "I am the servant of the servant" is that they are projecting themself lower than the other devotee (even though they are themself a devotee) and saying "I want to get closer to Krishna, and if you are a Servant in the House of Krishna, then if I become your servant than perhaps I can get closer to Krishna since I am in love with Krishna".
Understand?
But ultimately, it is about love.
Now to get to the crux of the matter of this love.
in these sects, love means Two. Meaning love cannot exist unless there is a minimum of two. The lover and the object of love. But in Prema Bhakti, this love or Prema can only exist if the devotee is SEPARATE from Krishna. Meaning they love Krishna, Krishna is the object, and the love itself can only exist if both are existing, i.e. Two.
This is not slavery. To this end, the love is so nice they want it to last forever, thus they do not want to be anything but a devotee, e.g. TWO EXIST.
"Can I serve you Krishna with sweet rice?", e.g. can I be before You and relish Your vision? "Oh, let me make a garland (for You)", another chance to relish. These are called Leelas.
Om Namah Sivaya