In HACKENBUSH: a window to a new world of math Owen says that ☽ (loony) is equal to the set of no nimbers (eg. $☽=\{\}$).
That seems to imply $\emptyset = ☽$. Is that the case? Or is $\emptyset || ☽$?
Additionally, in the video Owen also says loony is "apparently used for something else too(?)" - but doesn't go on to say what or give any references. What else is ☽ used for?
Overall meaning
In all uses I am aware of, ☽ represents a position that you would be loony to move to (typically in a component of a larger game), because you would lose the game, or at least gain no benefit from the move.
The details of this vary a bit depending on the context.
Types of uses
Use #1
Context
In the second edition of Winning Ways, Chapter 12 "Games Eternal - Games Entailed" has a relevant section called "Entailing Moves", which starts on page 396 (which I found by looking up "loony" in the index). In this section, it is argued that best play under normal play in an impartial game that would be a disjunctive sum with a special rule about "entailing moves" which require a response in the same component can be understood by assigning not just a single nimber to each component (as in the Sprague-Grundy theory), but rather a set of nimbers.
Definition
In this context, the empty set of nimbers happens to be what you would assign to a position where moving to it loses the entire game. As such, they define ☽ on page 398 to be the empty set of no nimbers. So here the answer to ☽$\,=\varnothing$ is technically "yes".
Caveat
However, I want to caution that this empty set is not the same "empty set" that you might think of as represented by the nimber $*0$ (or the game $0$ in a partizan/surreal context). The nimber $*0$ when treated as a set is "the set of positions you can move to from the canonical game with Grundy value $0$", which happens to be $\varnothing$.
But 1. other impartial games treated as sets also have the Grundy value $0$ and 2. the sets in the discussion of games with entailing moves (where ☽ arises) are not the sets of positions you can move to, nor even the sets of nimbers/Grundy values of positions you can move to (because in this context, positions don't typically have meaningful single Grundy values).
Use #2
On page 322 in Chapter 10 of the second edition of Winning Ways, there is a table of "tallies" for the game "Baked Alaska", in which they are treating square cakes as illegal positions. They write "we've...written simply ☽ for the illegal square cakes". However, the initial description of Baked Alaska says that a person who sees a square cake may win instantly. This means that you would have to be loony to move to them.
Here, no definition about ☽ being any particular set is made, because there is no need. But it is still like "Use #1" in that a move to ☽ loses the game (and so you would have to be loony to move to ☽ in a component).
Since Owen Maitzen seemed to primarily be using Winning Ways as his source, I suspect this was the "something else" referred to in the video.
Use #3
In "The Dots and Boxes Game" by Elwyn Berlekamp, he defines a "loony move" (to a "loony position") as one in which a certain general theorem guarantees that the opponent will be able to score at least half of the remaining points in the game. He also uses the ☽ symbol to denote positions that are not worth moving to (probably because they are loony positions in this sense).
Here again, there is no natural set definition for ☽.
Bonus question
Remaining is the question as to whether $\varnothing\parallel$☽. However, $\varnothing$ is not the way any partizan game is written (so that $\parallel$ is not a symbol that usually applies). And if we are in an impartial context so that $\varnothing$ represents the impartial game with no moves available, then $\parallel$ just means $\ne$, in which case $\varnothing\ne$☽ is indeed true in the context of Use # 1, and I suppose could depend on the play convention in a more general context.