Betteridge's law of headlines doesn't exactly fit here, as the answer is a kinda... maybe.
Yes, as of June 1st for the most part, people who use Google Photos to use the unlimited storage backup will be no more. Regardless of what option you use to back up option you use (High Quality or Original) for many people those photos will now start to count against your Google storage limit (many people are automatically gifted with the basic 15GB amount unless you pay for more).
But, for some there is an option. Essentially, the differential will be are you using a Pixel or not. If you're not, just forget about this and learn to start living within your storage limit (they do give you a calculator to determine how long it will last); if you're on a Pixel, it depends on which Pixel you're using to see what your option is, check out my handy dandy chart below to see where you may fall.
I have to say that regardless of where you sit in all this with your phone and Google Drive plans, it just plain SUCKS. One of THE features of Google I would always tell people about is that this feature allowed you to have that unlimited backup. I threw EVERYTHING I had from old photos in there. Years worth and it was just awesome to have that archive in there and then Google's intelligent features of finding faces and things; it was great for organizations to have that space.
Okay, here's my little chart for pre June 1st and post.
My take away from this is if you have an original Pixel, you need to keep that phone a rollin' and save that phone to do all your uploading. Copy anything to that phone and upload like crazy.
My work iPhone falls in that last category, so will lose the opportunity for unlimited uploads at high quality... same as my Pixelbook. Sigh.