When you buy things, you might have a purpose in mind for which they would be used.
It would be more natural to say:
On the way to the subway station, I buy some bread for my breakfast.
On the way to the subway station, I buy some bread for breakfast.
It is understood that it is your breakfast so you don't have to modify it with "my".
It is usual to say
have breakfast. One doesn't say usually say
have bread such as "have bread on the subway", but one would say 'have bread for breakfast'. It is more usual to have a soda or a drink, but one doesn't say "have a bread". One usually eats bread.
Breakfast does somewhat imply sitting down for a meal. In the context that you are describing, I wouldn't say you are having breakfast, but rather,
wolfing down breakfast as is common in the American way.
It is correct point out it is natural to use a construction like "I have bread all the time." or "I have bread everyday." or "I have bread in the refrigerator." Sitting at a table, it seems natural to ask a guest "Would you care for some bread?" Or "Have a slice of bread?"