supernova - What could be a star-like object that shines at daytime for few seconds? What's the probability to see it?

Several satellites (but most notably Iridium) have large reflective panels. If aligned with the sun properly, it can shine with sufficient brightness to be visible in the daytime.

An Iridium Flare usually lasts only a few seconds. It sounds quite consistent with your description.

It can be hard to tell in a blue sky with no nearby references, but an Iridium satellite would also be moving while fading out.

There are sites that can show you upcoming flares for your area. If you knew the exact date in the past, they could probably even "post-dict" what flares were available then and see if it matched.