It isn't false, it's a DOJ opinion. The option says that to initiate an impeachment inquiry the House must vote and otherwise it's invalid. They didn't. The President can deny subpoenas based on this opinion.
But that opinion is wrong; like Engel's opinion on McGahn's immunity from having to testify to Congress, and Engel's opinion on not sending the Trump-Ukraine extortion whistleblower complaint to Congress, and pretty much every other legal opinion Engel wrote because his job was to come up with bullshit legal justifications for whatever the Trump Admin wanted.
They won't use his opinion because they don't support it, agree with it, or think it'll stand up to any judicial scrutiny. And they sure don't want to set the expectation that they're going to bind themselves to the legal opinions Trump's cavalcade of clown lawyers pulled out of their asses.
These “opinions,” even if they were legally sound to start (which this one wasn’t), are not binding on anybody, especially subsequent administrations. The article is also based on the ludicrously false premise that Republicans expect the rules that apply to them to apply equally to Democrats, and that a precedent favorable to their position must be extended to non-fascists as well. We need look no further than the current head of the DOJ to see that’s not true.