I think 3D printing would have a lot of potential application making toys and accessories. (E.g. those little T-fittings some cages are put together with, if you lose one...)
I built my own 3D printer from a kit in 2012 so I thought this thread was a neat idea. Sorry if it's too old to reply to, just wanted to get my two cents in here since I'm enthusiastic about 3D printing in general.
However, I can also see a few problems here. I would definitely keep a bird and a 3D printer well apart. Many filament-based 3D printers have PTFE (Teflon) in their 'hot end'. Normally it won't get so hot that the PTFE decomposes, but malfunctions happen and there is a fairly large amount of PTFE in some hot end designs, probably multiple non-stick pans worth. You can get all-metal hot ends with no PTFE, but some plastics you can print with also put off tiny particles and fumes when they are melted. Resin-based printers on the other hand do not have hot parts, but they do use solvents and resins that may have volatile components that could compromise indoor air quality as well.
Also, filament-printed objects often have lined/grooved surface texture on 4 of their six sides, which is probably great for holding onto with your claws, but could also harbor bacteria and be difficult to clean when effective liquid cleaners like bleach are unavailable. (Resin printers do not really have this problem but printing large resin objects is pretty expensive. Printing a perch out of resin would not be cost effective, unless maybe you keep hummingbirds!)
If you steam clean things, some of the common 3D printing plastics will be damaged by it. For example, the most common inexpensive filament is PLA, which has a softening point of around 65C, so the 100+C steam coming from the steam cleaner will make it get soft and maybe distort its shape under its own weight. Modern filament-based printers can print in polycarbonate, which is very strong and resilient and is nontoxic and softens only well above the temperature of boiling water. That might be a good choice for a material.