“マイク は このよう な（この様 な） お客様 と 話し を する 時 幸せに 感じます 。”
what is the function of と in that context (お客様 と 話し を) ?
It means “with”. Mike’s happy when he speaks with such customers.
Notice that you have the exact same type of “と” in the previous sentence:
Mike can meet [with, needed in Japanese, although not in English] a lot of friendly people.
By the way, this to = with is the same as the one that in other contexts gets translated as “and”, they function as particles that attach to nouns, equivalent to English prepositions (such as “with”):.
It’s different from the “quoting” and the conditional “to”, both of which attach to clauses and are thus, conjunctions.