I have two explanations why this is the case:
Although Silver Chariot has gone Berserk, it's still his Stand
First of all, although he can't control Silver Chariot Requiem anymore doesn't mean it's not his Stand anymore. In the flashback where Polnareff first found out about what happens when Silver Chariot has the Stand-Arrow, we can see that after taking the Arrow from Silver Chariot Requiem away again, the Stand transforms back into it's former normal self. So I'd say that even in the case of Polnareff not being able to control his Requiem-Stand anymore, he still is a Stand-User. Also, the reason why he was able to pick the Arrow up was because his Stand was already active and the effect of the Arrow (your Stand attacking you) still happens as we can see with Silver Chariot Requiem running towards him after picking up the Arrow.
Polnareff took over the Turtle's Stand
The second case why he can still see Stands is because the Turtle (aka. Coco Jumbo) was a Stand-User and Polnareff's soul got transferred into the Turtle. At the end we also see that Coco Jumbo's Stand "Mr. President" is still active because we see Trish getting transferred inside the Turtle just like before the Silver Chariot Requiem incident. I'm guessing that Polnareff just took over the Turtle's Stand which is definitely plausible in certain cases.
Part 6 Spoiler:
Obtaining someone else's Stand in addition to your own is possible in some special cases like we see with Emporio being able to use Weather Report in addition to Burning down the House at the end of Stone Ocean
So although it isn't explicitly stated in the anime/manga, Polnareff still certainly is a Stand-User.