Canada has a variation on this law.
http://laws-lois.justice.gc.ca/eng/acts/R-4.5/page-1.html#h-...
Edit: Also note that in the preamble it specifically says "Canadians and small businesses." This law does not impact, for example, environmental or safety regulations for real estate developers or utilities. Trump will almost certainly try to cut EPA regulations but something like the Canadian law would never allow that.
You're pointing to a proposal (or more succinctly an initiative). Saying it is nothing like an implementation is misleading.
Regulations are issued by the executive branch, not Congress (repealing regulations is actually done by issuing regulations specifying the regulations to be repealed, so the process for repeal is the same.)
Also, "a regulation" isn't a well-defined unit.
Sounds like the meaning of the executive's order to 'repeal two for every one new' could be decided within the executive branch, then. And since everyone within the executive branch is responsible to the chief executive … maybe it could even work.
He can certainly try: http://www.npr.org/sections/thetwo-way/2014/11/20/365519963/...
In the general case, though, a lot of legislation gives the executive wide discretion as to the details of regulations, and so the president can direct his subordinates to change regulations however he wants, within what the law allows.
E.g. with a budget example the Congress might want to subsidise cantaloupe farms, and budget $100,000,000 for cantaloupe farmers, to be allocated 'as the Secretary of Agriculture may direct.' The president would be within his rights to order the Secretary of Agriculture to only allocate that money to cantaloupe farmers with a total net worth of under $1,000,000, or to give no more than $100,000 to any particular cantaloupe farmer, or (possibly — this one's a stretch) to prioritise farmers of the famed Golden Lucy cantaloupe before all others. But he can't direct that the money be spent on rutabaga farmers.
Similarly, if the Congress writes a law which gives the pertinent executive department some latitude, the President may order that department to exercise that latitude however he likes.