They don’t have exactly the same meaning, generally. ‘Starting out’ = first thing you do ‘Starting off’ is usually more purposeful. The marathon’s in 4 months. Start off by setting up a training calendar and going to a good running store where they can analyze your gait. (The first things you do have a purpose) When I first started out in this business, I didn’t know what the hell I was doing. I learned from my mistakes, though. (They were the first things I did, but weren’t well planned) Consider a physical journey: I started out on June the 18th and finished 27 days later. (I departed) I started off by going over the Pyrenees into Spain. (I accomplished something)

To understand prepositions, always start off with their physical meanings. All their metaphorical meanings derive from their physical meanings. If object=A is on a table=B, we say A is "on" B. We can then move A "off" of B, so it is no longer on the table. You cannot move A out of B because A is not inside of B. You can only move out of something that you are inside of. In your two examples, you start neither inside nor on top of anything. Therefore "off" and "out" are interchangeable in both sentences and it doesn't matter which one you use. The images feel different, however. When you say "start out your day", the image is of being released from being inside of something. It feels like you are "going forth" as you exit your present circumstances. When you "start off your day", the image is of being pushed off or jumping off of something, like a racer who has just crossed the starting line.

The context they are used in most of the time can be used interchangeably...Meaning they similar

I think generally speaking that have the same meaning. It might depend on the context, but in the examples you have given they have a similar meaning. Hope this help! Phrasal verbs are tricky! ;)