I have a 3 conductor (Line1, Line2, ground) (240V) wire that goes to a step-down transformer to get 120V. Since the transformer output is 120V between its line1 and line2 and there is no neutral, does this affect what kinds of devices can be plugged in? Hope I explained that clearly enough...
You said "Step-down transformer", not "isolation transformer".
The commonly available step-down transformers are autotransformers. That means they only have a primary winding, no secondary at all. 120V is picked up at the center tap. 240V is applied to the top and bottom taps, and 120V is taken off the center and bottom taps.
Now, typically the bottom tap is considered to be "Neutral". But in North American style (and Philippines) 240V power, both legs of 240V are not anywhere near neutral, to be more precise, they are ~120V from neutral. That means the center-tap of this transformer will be "somewhere in the neighborhood of ground (and where neutral would be)", but not close enough to tie them together - and anyway, that would violate the sacrosanct Only One Ground Bond rule.
Note that the bottom tap is "considered neutral" in these step-down transformers, but the middle tap is actually the one near ground. So the polarity would be opposite what you'd expect, and I'd expect it to light "Hot-Neutral Reverse" on a magic-8-ball tester (this being more or less true).
The upshot is you should use double-insulated devices, which don't care about either wire being neutral.
If you wanted to do this properly, you would use an isolation transformer, treat the output of its secondary as a separate main service... establish ground via local ground rods, and pick an output wire to equipotential-bond to ground (to prevent floating). Its name would be Neutral.
Assuming you have a traditional, isolated transformer with separate primary and secondary windings, the output of your transformer as currently configured is floating, meaning that while there is 120V between line1 and line2, the voltage from either of those to ground could be anything, and can change over time. If you want to make it a traditional hot/neutral setup, you can bond one of the two output lines to ground. If you're wiring this as the input into a breaker panel, make the bond there -- this makes it a main service panel, from a separately derived service, rather than a subpanel. The bonded lead then becomes your neutral, and the other is your 120V hot.
If instead your transformer is really an autotransformer, where the primary and secondary windings are actually the same thing, then you cannot ground one of the outputs. In this case, the voltage is not floating, but where exactly it is depends on the transformer's construction. I'd recommend measuring from each of the hot leads to ground to see if one of them is at 0V already, or if they are each approximately 60V from ground, or something else.
To check if it's an autotransformer or a regular transformer, check if there's any continuity between the primary windings (input) and secondary windings (output). If there is, then it's an autotransformer.