So far, Odin's serialization has held up very well since we released, and has consistently proven to be the most reliable, bug-free part of the plugin. It was also the part of Odin that was written first, and so it saw heavy use and testing during the development of the rest of Odin. Our cautionary warnings are merely us exercising due caution.
A lot rests on serialization. As such, we want to make sure it is 100% safe to use before recommending people to use it in their projects. We're confident Odin's serialization feature currently works as expected; however, even a small issue could theoretically corrupt a project's serialized data quickly. We haven't had any such issues, but with people's precious serialized data, there is no such thing as being too safe. Therefore we want to very, very thoroughly vet and test Odin's serialization before we actively recommend people to use it, and that seems to be going very well.
Additionally, as is the main point we state in our manual, the freedom that Odin's serialization gives you to arrange your data exactly as you want it can be a mixed blessing. On the positive side, you can do anything you want, and on the negative side, well, you can do anything you want. More freedom isn't always good, unless it comes with proportionally more restraint. It is very easy for people to get themselves into a mess with highly complex data structures that have no real need to be so complex, and could be better expressed in a simpler way.
Therefore, as a rule we recommend people to exercise due caution and restraint when using our serialization system, as we would with any other tool that's so powerful and flexible. It's always best to let Unity serialize your data if you can, since Unity's system is simpler, faster, and has been in use for much longer. Odin's serialization is intended to lap the gaping holes in Unity's serialization that people find so painful - not to replace it entirely.
In terms of future compatibility-breaking changes, it's very simple: they won't happen. It is very important to us that the serializer is and remains totally backwards-compatible. The robustness and integrity of the serialization system is, and remains, our first priority.
So, to summarize: Can you use Odin's serializer? Absolutely, but be aware of the risks - both on your end and on the serialization's - and always make sure to keep backups and use source control software, just in case. After all, that's just common sense.