In 2010, the Republican Party won control of the House of Representatives, but not the Senate (and obviously, not the White House). They decided that their method of winning policy fights would be to take hostages. Mitch McConnell even admitted as much, referring to the debt ceiling as, "a hostage that's worth ransoming." This was nuts. The Treasury needed the authority to issue bonds in order to cover the spending that Congress had already ordered, and without the authority to do so, we risked worldwide financial calamity. Republicans were demanding policy concessions in order to not tank the economy of every country on the planet. At least both things related to the spending, though... Obama caved in 2011 and gave the GOP the 2011 Budget Control Act, but eventually he wised up on the debt ceiling and stopped paying ransom.
In 2013, though, Senate pied piper Ted Cruz led a bunch of House rats (the Freedom Caucus, mostly) to their demise. He told them that they should demand a government shutdown in exchange for a delay in implementation of Obamacare, or possibly defunding, or possibly, something, 'cuz... OBAMACARE!!!! Yeah, that didn't work out so well. The GOP had to cave. Boehner and McConnell knew that would be the result. They always hated Ted Cruz. The point, for Cruz, was always that Boehner and McConnell would cave, and that Cruz would be able to call them cuckservatives for doing so. He really is a shitbag. This is kind of the critical story, though, because the 2013 shutdown failed spectacularly. The GOP was demanding completely unrelated concessions in exchange for the continued operation of government. It made them look like assholes. The way you win a shutdown is by making the other side look like assholes. That's hard to do when Ted Cruz is leading your side's charge... That's also hard to do when your demand is unrelated to the primary issue at hand-- the continued operation of government.
So, here we are. The House passed a continuing resolution. Why a continuing resolution? Because actual appropriations bills are soooo 20th century! Whether or not we get a shutdown now depends on the Senate, and specifically, Senate Democrats. PredictIt right now puts the odds at just below a coin toss. The problem is that right now, their demand is DACA, and DACA is dead. More than that, it is not germane to the shutdown itself.
If the Senate Democrats shut down the government over DACA, they are pulling a Ted Cruz. Will they? Odds right now are slightly against it, but at this point, things are uncertain. If they do go for a shutdown, what happens? Part of Ted Cruz's problem was that he is Ted Cruz. It is hard to look like the bad guy when your adversary is Ted Cruz.
But you know who managed that task? Trump. By calling Cruz's wife ugly and accusing his father of participating in the Kennedy assassination. Of course, Ted Cruz managed to undercut any sympathy anyone might have for him by allowing himself to be brought to heel, but this sort of makes an important point about the dynamic here and the underlying question.
Yes, if the Senate Democrats shut down the government over DACA, they are doing what Republicans used to do. Same fuckin' thing. Hostage-taking. The thing that puts them in a somewhat better position is that their adversary is... Donald Trump.
Remember, though, that in 1995, when Newt Gingrich shut down the government, Bill Clinton was considered politically inept. His first two years consisted of nothing but bumbling. A lot of people look back on Clinton as a political operator of rare gift, but in 1995, that wasn't what people thought of him.
Will Trump show some heretofore unseen stable-genius-ness? Ummm... not likely. That isn't what the Montreal Cognitive Assessment assessed. However, don't assume you know how this plays out. If the Senate Democrats shut down the government here, they are the Ted Cruz of this situation, regardless of what you think of the policies they are pursuing, and Gingrich wrote off Clinton's ability to beat him.
Right now, I have no idea how this plays out.