The speaker draws a line between two kinds of company. With a dog, there's no need to pretend. The only "alteration" is play, so the speaker can simply be silly, present, and unguarded. A dog doesn't care how old you are or how you're supposed to behave.
With other people, things change. They're described as "weary / as office business," a neat comparison that makes adult company feel tired and routine. For them, the speaker "plays the age tune," as if age were a song everyone expects to hear. The speaker goes along with it and "keeps time," which suggests both rhythm and the passing of years.
The mood is quiet, wry, and a little wistful. There's no anger here, just a gentle awareness that acting your age is a performance, while playing with a dog is honest. The word "play" does double duty, meaning both fun and acting a part.