Not exactly. For instance a graduate of an accredited PA program can take their national certification exam up to 6 years/6 attempts post graduation. While being NCCPA certified could be, possibly be, considered as "board certified", it is not exactly the same as MD board certification. For MDs, board certification comes after taking USMLE, completing advanced training (residency), and sitting for a specialty exam. The USMLE is more comparable to what PAs and NPs do. Not in content/depth/structure, but as in we all must pass these exams before we can move on to the next step post finishing school. For MD it is residency (although they can take step III before residency or after being accepted into residency) and for NP/PA it is generally licensure and working. The term "boards" amongst PAs and NPs is often used as in passing a national certification exam, but not the same as being board certified/eligible physician. As a PA/NP you can't gain licensure* or a job just because your are eligible to sit for the national certification exam.
The statement board certified/board eligible mostly relates to physicians who have completed residency, but one could argue that PA/NP can use the term board certified to mean they passed a national certification exam.
*there may be some states that still allow for temporary licensure after graduation, but before becoming certified for PAs and perhaps a caveat to this in regard to NP as well.