The phrase "becomes the same" is correct and usable in written English.
It is used to indicate a change from one state to an identical one. For example, "As the seasons change from winter to spring, the temperature slowly becomes the same."
The height of her letters becomes the same as the thickness of her pen nib.
"When everything becomes the same because of winemaking practices, that's a pretty sad day".
From the minute he's gone it becomes the same mess it is today.
When you have corporate greed come in, everything becomes the same, and I find that very boring.
Los Angeles becomes the same sunny city Mr. Wilson defined with the Beach Boys in the mid-1960s, now tinged with regrets for lost time.
The Congressional Research Service in a memorandum this week said the number "normally becomes the same as a majority of the number of electoral votes counted".
"If I'm listening to something on the radio and it becomes the same thing over and over," Mr. Haynes said last week, while sitting in the living room of his house near Long Island's south shore, "I change the channel.
Any potential attempt to collude with a hostile foreign government to undermine democracy becomes the same as an attempt by others to find out if anyone has been potentially colluding.
At that point our collective imaginations become one imagination and my internal life becomes the same as your internal life, which is what Aristotle understood when he analyzed tragedy.
In the absence of such a membrane, the solvent and every species of dissolved ion will diffuse freely from each solution into the other, until the composition of the two solutions becomes the same.
