Welcome. You’re in the perfect place for this and it’s a great question.
If the psd file you got actually has an embedded color profile in it, then you actually do have the profile - it’s embedded in the image file and it will be used by Photoshop to view your image to you. Unfortunately this embedded profile you got is most likely a default monitor profile that is created by the operating system to define in a general way your client’s monitor. Usually the purpose of embedding a profile is to identify the working space the client is in - something like AdobeRGB or sRGB. It’s not very helpful to include a monitor profile. I would recommend that you “un-assign” the profile in Photoshop so that you’re not using that one.
Check out color management myth #7:
colorwiki.com/wiki/Color_Man … Myths_6-10
How do you know the colors are not right when you are exporting to the web? Do the colors desaturate? If so, then my guess is that you are converting a well saturated image from AdobeRGB to sRGB and that will necessarily involve some desaturation.
As far as Mac vs. Win: Color management is made to be platform independent. In principle, you should have the same results whether you are working on a Mac or a Windows machine. In practice there are a lot of settings and a lot of different elements of a workflow that might explain why you are not getting the color that you expect.