Drano crystals is just sodium hydroxide (lye) with aluminum chips. There may be slight amount of other stuff in there, but it is mostly lye. When wet, the lye dissolves, eats the aluminum, creating a lot of heat and bubbles. The Liquid Drano (and similar products) are caustic solutions that have been jelled to help them “pour through standing water and get to the clog”. Very similar to the crystals, but there isn’t any aluminum and the water is already added. When used as directed, the caustic should be safe for pipes.
I can believe engineer_comp_geek’s anecdote about it eating pipes. I have seen it. Nickel-Plated brass pipes. These were very popular P-traps about 50 years ago. Lye will corrode the brass, but it takes a long time. The directions for Drano call for allowing to sit for 30 minutes and flushing with cold water for 15 minutes afterwards. If I had to hazard a guess, I’d say that in his example, the Drano was added in the evening and allowed to sit overnight before flushing, regularly. This will corrode the brass pipe.
Caustic drain cleaners will clear hair clogs (common in the bathroom) by dissolving the proteins. Strong caustics do a good job of dissolving proteins, which is why they are so dangerous to handle, as your skin is also made of proteins (as is just about everything else in your body) and dissolving them is not generally a good idea.
Caustic drain cleaners will also clear grease clogs (common in the kitchen), by saponifying the oils (turns them to soap).
The acidic drain cleaners I have dealt with are sulfuric acid based. While dangerous to handle, I am less concerned about strong acids than strong caustics. Both are bad, but the caustics are generally thicker and are more difficult to wash away. Both will create fumes that will irritate your eyes and nose, but the casutic will blind you if if gets into your eyes. The acid will, too, but you have a chance to save your sight if you wash it out quickly enough. Not so with the caustic. Strong acid is bad. Strong caustic is worse.
So, to answer the OP, yes, Drano should be be safe if you use it as directed. I would not use it if the drain is completely clogged. If it just drains slowly, it should be OK. Be sure to rinse well after the recommended rest time and there should be no lasting damage. Once diluted, lye (sodium hydroxide) poses no environmental danger.