Yes, whenever possible, full name, company, and link is the best way to go for all the reasons listed already.
There can be exceptions. It might depend on the work you did. I imagine if your clients are, say, lawyers or medical professionals, they might sometimes need to stay anonymous for legal or privacy reasons.
I have a mixture of full and partial IDs in my testimonials. One of my services is ghostwriting, and ghostwriting clients often don't want it widely known that they didn't do their own writing. That's just an accepted part of the ghosting deal. But some will compromise. I have one ghostwriting client who didn't want his or his company's name listed on the Web for the whole world to see. But he is willing for me to share the info privately, by email or phone, if a prospect asks for a reference. I also have a testimonial from someone who no longer works for the company she did when I worked with her, and she left that job on unpleasant terms, so she didn't want to list either her full name OR the company's link for both their sakes. But she still gave me a very nice quote, which I attribute by her first name and former position title only, and she also gave me permission to share her contact info privately.
How's that for a long answer to a short question?