Is it appropriate to apply for assistant professor positions a few months before obtaining my PhD?
In most biology/biological sciences departments in the US and Canada it would be unusual for someone to be hired into a tenure-track assistant professorship immediately following their PhD; most candidates would have at least a year or two of postdoctoral experience. (This may be different in biomedical and/or clinically oriented programs). If you have a spectacular research record that could make you an exception.
Postdoctoral experience is less expected in math (although 1-2 years is probably still the median), and even less in statistics; this is probably also the case in other fields such as engineering where people with a fresh PhD can get well-paying jobs in industry.
It doesn't hurt to apply for such positions, but I would say if would be safer to at least consider a few post-doctoral positions as a backup plan.
Finally, as with most questions like this, you will get the most relevant advice from your supervisor and colleagues in your lab/program; they know the most about the norms of your particular sub-field and geographic region. Unless you have some particular reason not to, you should probably ask them.

Here is some data from a reasonably recent survey of (mostly North American) ecologists (note, this paper was contentious because of the way it handled gender and informed consent of respondents):

Recently hired TT assistant professors of ecology typically were about 4 years post-Ph.D. at the time of hiring (mean 4.2 years, median 4). 69% had anywhere from 2 to 6 years of post-Ph.D. experience, with a range from 0 to 11 years of post-Ph.D. experience. The majority of ecology faculty job seekers also are 2–6 years post-Ph.D.


These data are consistent with a job market in which most hiring institutions prefer applicants with at least a year or two of post-Ph.D. experience. But once you have a few years of post-Ph.D. experience, the marginal value of additional post-Ph.D. experience appears to be low at best, though it's hard to say precisely based on the available data.

Fox, Jeremy. “A Data-Based Guide to the North American Ecology Faculty Job Market.” The Bulletin of the Ecological Society of America 101, no. 2 (2020): e01624. https://doi.org/10.1002/bes2.1624.