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.
or simply
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?"