What is the significance of a new name being given in the Bhakti Practice?

Yoga is the effort to establish a true identity, rather than an imposed identity (YS 1.2-3). One’s name has a huge influence on one’s identity. So, a name that has some spiritual reference is helpful in any form of yoga.

Where does the name come from – the devotee themselves or their teacher?

Usually, it’s from the teacher – similar to how a parent names a child. When it is done that way, then you get the additional desirable impact on identity of being related to a particular teacher that inspires you. But there is no reason it couldn’t also be self-assigned.

Is it supposed to be a way of designating an individual who has been accepted by a guru?

It does seem to often be used that way, which I think may be more harmful than helpful to spiritual progress. In any case, that’s not the original or primary intent; which was simply to propel the identity evolution.

In Kali Yuga should we even want such a name, or just continue with our birth-name.

Honestly, we should be aware that the goal of yoga is to establish true-identity, not imposed identity, and that there is absolutely no guarantee that a spiritual name will help instead of just imposing more extrinsic identity. In the majority of cases it just becomes an ornament of pride, or superiority, or it eventually becomes laced with some distaste due to being associated with a teacher or group that we no longer hold in esteem. So, I would just say, “its not a big deal.” There are way bigger and more important things to think about and focus on in Bhakti-Yoga. It’s cool if you wind up with a name, and I think it eventually happens naturally and beautifully if you are doing things sincerely, but it’s not something to go on a quest for.