It maybe a feature that's unique to Samsung Galaxy S5 OS 6.0.
Update: Also, to further explain, the favorite contacts aren't actually labeled "Favorites" on the Share Bar, but rather the list of contacts are simply designated as such in the Contacts application. One of the contacts in the list is someone that I rarely send mail or texts to, so I know it's not based on the frequency of sharing with or sending to a particular contact. In other words, the list of contacts on the Share Bar matches the list of Contacts that are marked as Favorites in my Contacts application.
See attached screenshot for an example of how this presents on the Share Bar...
I just looked at another person's Samsung Galaxy S5 with Android 6.0 on it, and there is nothing like that there. I am not sure if that person actually has something under "Favorites" in Contacts or not (probably not). So, in that "Share bar", there is Aquamail, but nothing else from it.
Based on your description and snapshot and on Kostya's response (which I understand better now), - I am guessing that when there are entries in the contacts that are marked as "Starred" (I just recalled that it can be done in my phone's "People" (stock contacts app), and I remember that Aquamail actually has an action for a message "forward to starred"), - then those contacts are offered as separate and direct options to share with them via Aquamail.
In this scenario, my earlier thoughts in this thread are incorrect, as there is probably nothing set as the "default" app for the particular sharing action. But then, I am confused by the statement that those "Favorites" have replaced "Verizon Message+". I am guessing that you probably mean tthat previously the same "Favorites" were listed under "Share", but with the default actions for them being Verizon Message+ instead of Aquamail. Right?
I guess, Kostya has some solution for that (adding a config option in Aquamail). But I would be curious how that same thing can be changed at the system settings level. It looks like one app can block another one and the user cannot change that... that's strange... Or, maybe there is something new for that in Android 6.x?