"My second thought is ,although in their eyes it is just a holiday greeting, I see it as an advertisement and we are selecting only a few mortgage customers to get this advertisement. Isn't that a problem?"
Like Randy mentioned, it meets the qualifications of an advertisement. The part that many bankers do not understand is that the BANK gets evaluated on marketing and advertising, not the loan officers. If all the advertising is to (for example) middle age white guys, that could be a problem. Same as if it went to mostly minorities (race, gender, ethnicity), encouraging them to refinance. Another problem. If the recipients are in non-minority upper income neighborhoods,, yes, you guessed it, a problem.
Some things tested for in Marketing during a fair lending exam
M1. Advertising patterns or practices that a reasonable person would believe indicate prohibited basis customers are less desirable.
M2. Advertising only in media serving particular racial or national origin areas of the market.
M4. Use of marketing programs or procedures for residential loan products that exclude one or more regions or geographies within the bank’s assessment or marketing area that have significantly higher percentages of residents of a particular racial or national origin group than does the remainder of the assessment or marketing area.
M6. *Proportion of monitored prohibited basis applicants is significantly lower than that group’s representation in the total population of the market area.
Integrity. With it, nothing else matters. Without it, nothing else matters.