You're using the present perfect tense in the first, and the present perfect progressive in the second.
present perfect tense
1. "We have eaten breakfast together every morning since our honeymoon."
This means that as a married couple, you and your spouse have never spent a single breakfast alone from the first day of your marriage to now, the present day. The action is repeated, and does not exclude the fact that having breakfast together requires a certain duration of time.
Perfectly acceptable, although an unlikely situation if the marriage is longer than six months. (I'm joking!)
Present perfect *progressive/continuous* tense
2. We have been eating breakfast together every morning since our honeymoon.
Literally, this could be interpreted as both of you eating breakfast non-stop since day one of your marriage.
This of course does not make sense; however, the fact you have inserted ever morning means the action is being repeated, it is not uninterrupted as it could appear at first glance. You need to avoid this ambiguity, native speakers will tend to do this automatically without thinking.
Opting for:
Present perfect
"We have eaten breakfast together every morning since our honeymoon."
or
Present perfect
"We have always had/eaten breakfast together since the day of our honeymoon."
or
Simple present
We always have breakfast together, ever since our honeymoon.
(Insert always to emphasize an action is repeated regularly).
Best Answer
Fairly familiar usage, ordered by how recently I ate, with common implications:
"I have finished eating" -- I just now finished eating.
"I am done eating" -- I recently finished eating; I will not eat again soon.
"I have eaten" -- I ate earlier today; I am not hungry now.