Mr. and Mrs. Smith are agents with The Factory.
That is a private security/intelligence corporation open to contracts with whomever they want, I would expect, but which seems to concentrate on working with the American government. It is run by a man we only know as Mr. Big. We do not learn a whole lot about the organization except that it is extremely well funded and has a world-wide presence.
We also know that they are not the only company of its type in operation for while Mr. Smith is an operative with The Factory from the first time we meet him, Mrs. Smith is employed by a rival of theirs. Her employment will be terminated and she will transition to The Factory soon thereafter.
"I'm Mr. Smith but that's not my real name," we are told by Mr. Smith in the standard introduction to each of the adventures. Mrs. Smith then quickly adds, "I'm Mrs. Smith but that's not my real name either."
"You see," he goes on to tells as if it comes as surprise, "we're spies. People might think that we're married."
"That's okay," she replies, "as long as you don't." Ouch!
The Factory apparently feels that a team of two is better for most operations than a lone wolf. Two men or two women traveling together might be rare enough to attract attention but a man and a woman pair is far more innocuous, especially if they go as a married couple. Hence the Mr. and Mrs. Smith even though they are not actually married.
If what the Smith's say can be believed (and they are spies, remember), they do not even know each other's real name. And they may or may not like each other (kinda depends on the day). And there is constantly a nice bit of rivalry between the two with both of these experience agents quite confident in his or her abilities and likely each thinking that person is just a tad bit better at their job than the other.