Not all browsers support colour management, and if they don't, they'll act like the images are encoded in sRGB. Since sRGB is also the default for images without an embedded profile, this is the safest option for images intended to be viewed via web browser.
It doesn't make much of a difference whether you embed sRGB or no profile at all.
PS does have an option in the colour management settings to warn you if an images colour profile differs from the set working profile - this is a good idea to turn on, then you'll see the images' colour profile in the warning dialog on opening the image, together with the option to convert it, use the images profile as working profile or ignore the difference (not recommended).
You can set the information displayed at the bottom of each image window in PS to show you the colour space.