
Can someone please explain why these two sentences are stressed differently? What does this difference achieve? Is it common?
Мы пойдём с конём по́ полю вдвоём
Мы пойдём с конём по по́лю вдвоём

Both variants are the same, "по по́лю" is more common, "по́ полю" sounds a little bit country-style.
But in this song these 2 styles are combined by another reason:
In Russian poetry, all sentences should have strict pattern: alternation of stressed and unstressed syllables should correspond to the meter of the poem.
Current song has such pattern, so that just this combination of stressed/unstressed syllables is possible.
And if, for instance, the phrase "по́ полю" did not exist in Russian, the double repetition of sentence "Мы пойдём с конём по по́лю вдвоём" would not possible here due to stress-pattern - and the author should use some another phrase.
But both variants exist, and author uses them.
This approach - repetition of phrases with different stress - sounds beautifully and is sometimes used in poetry.
another rendition:
https://www.youtube.com/watch?v=rcyriz-aolw