The president did take ownership of the bill.
His statement:
Remarks by President Trump on the Health Care Bill
“Thank you very much. We were very close, and it was a very, very tight margin. We had no Democrat support. We had no votes from the Democrats. They weren’t going to give us a single vote, so it’s a very difficult thing to do.”
https://www.whitehouse.gov/the-press-office/2017/03/24/remarks-president-trump-health-care-bill
Notice the president’s use of the word “we” and not “they,” the members of Congress.
It’s still not his bill it originated in the House and it belongs to the House, more specifically Paul Ryan. It would only be fairly described as Trump’s if he signed it into law.