My father used to say, (1)
Could there be a figure who more directly represents "Tradition and Customs" than a father? Your own father might be really cool and hip, but fathers in general are the universal symbol for "old school and old rules." Also, we're going to hear what the speaker's father used to say – what he said in the past, but doesn't say anymore. The speaker omits why he doesn't anymore, but we guess that the speaker wants to emphasize that her father and his values belong to the past.
have to be shown Longfellow's gravenor the glass flowers at Harvard. (3-4)
Both Longfellow's grave and the Harvard glass flowers represent 19th-century culture. Longfellow is definitely a 19th-century poet; his poetry basically displays everything that the modernists later overthrew, such as rhyming lines, a clear plot, and sentimental expressions. As for the glass flowers: although they were completed in the 1930s, they were begun in the 1880s and are strongly associated with a widespread 19th-century interest in natural history.
Self-reliant like the cat – that takes its prey to privacy,the mouse's limp tail hanging like a shoelace from its mouth – (5-7)
Literary scholar Elizabeth Gregory has discussed how these lines allude to Ralph Waldo Emerson, the 19th-century American writer and philosopher. You may have heard of him as the primary formulator of a branch of philosophy called Transcendentalism…but we don't really need to know that for Moore's poem. Emerson did, however, write an important essay called "Self-Reliance," and the image of a cat going after its own tail appears in his essay on "Experience." (Notice how Moore changes this to a mouse's tail, making the cat become a real predator.) So the speaker's father doesn't just make obvious references to 19th-century culture, he also embeds these references within his own words.