I think it's OK. I would NOT criticize your colleague. First, and most important, it communicates clearly, it's natural, and it's good idiomatic English. It is obviously conversational, spoken English. It is used in conversation to communicate a plan, confirm an understanding. I think it is professional. In clothing, there is a degree of formality in between a business suit and casual dress, called "business casual," and I think this is the language equivalent of "business casual."
I can't think of any shorter or more polite way to say "am I understanding you?" or "is that what you meant?" And I can't think of anything more professional than to make sure you have understood by repeating back something and saying "right?"
I'm a little old-fashioned, and if a travel agent used my first name in talking to me, I wouldn't exactly be offended but I would notice it and think it was a little out of place. I would NOT feel that way if a travel agent said something like "So, you would prefer Monday but if I can't find anything else Tuesday is your second choice, right?"
What, exactly, would you use instead?
It could be an irritating speech mannerism if somebody used it at the end of every single sentence.