I don't think anyone owns the shutdown per se. Everyone knew long before the shutdown began that Trump said he wasn't going to sign or approve another spending bill without funding for the wall. There was plenty of time for both sides to work out a compromise. Neither choose to do so. Both sides now have become entrench in either all the funding for the wall or no funding at all. No compromise is available at this time that wouldn't look like each side caved into the other. At least to each sides avid supporters, to the very anti and pro Trumpers.
This battle has turned very personal, a battle of ego's. It's not like when our government spends 4 trillion dollars a year, that a battle over 5 billion is worth all the heartaches and problems that the shutdown is occurring. In the end I would wager the shutdown has cost the government much more than just 5 billion. But this is looking at the problem through dollar signs and not ideology, not political partisanship, not being either pro or anti Trump, not being a Republican or a Democrat. Just hard financial facts.
This battle has become more important to attach blame, your OP. Than to solve the problem. Right now neither side is willing to solve the problem without total victory, unconditional surrender. Each side is sticking to their guns and their positions are written in quick drying cement. Right now 39% of Americans state they're for the wall, 46% oppose. Question 18.
https://d25d2506sfb94s.cloudfront.net/cumulus_uploads/document/f0al8raalq/econTabReport.pdf
Most to blame for the shutdown, that goes to Trump, 50%, Democrats 32%. Question 35. But if we reword the question to How much do you blame each of the groups listed below for causing the shutdown of the federal government? Democrats in congress, 40% blame them a lot, 21% blame then a little. Total 61%. Republicans in congress, 40% blame them a lot, 35% a little, total 75%. Trump, 56% blame him a lot, 16% a little, total 72%. Questions 34A, B, C.
My bottom line on all of this, it has become more important to attach blame for the shutdown, to make the other side cave 100% than to solve the problem and reopen the government. Neither side in my opinion is taking the damage to this nation and its cost into consideration. This has become nothing more than a knock down, drawn out fight between partisanship, ideology, pro and anti Trumpers. One side or the other will probably score a political victory, but the question will be in the end, was it worth it to the nation as a whole? The damage done may outweigh the price of victory.