I started my second postdoc at a European university this month and found that my department has a job posting for hiring an assistant professor with a tenure track. It matches my profile and I really wish to have a try. But I am not sure if my current supervisor would be unhappy about it and if my application would be considered by the department, because its application deadline is quite close and it would be just at the third week of my two-year postdoc contract...
A reasonable and supportive supervisor would have no trouble with this. You need to have a chat with them to see how disruptive it would be, but, I assume, that any permanent position wouldn't start for a while in any case.
In an ideal world, yes, this is an appropriate action. The purpose of a postdoc is to get a permanent position. There is nothing inappropriate in applying for one. Normally you need to be making applications during a postdoc in any case. This is just a bit early.
But a chat would be appropriate also.