Muslims do not get some kind of documentation when they become muslim, though some countries' passports might state religion.
So, in general, when encountering a 'Muslims only' sign, stating you are a muslim should be sufficient. If you're not, and don't know basic tenets, you might be frowned upon, or angrily removed from the premises.
I was born in Iran (to a technically muslim father) and this meant I am muslim by default, even if I don't practice. I have visited several places that were 'Muslims only' (outside of Iran) and, as I don't speak, act and dress like someone who grew up in Iran, have sometimes been questioned. I simply would state I'm muslim and that was that, every time.
However, the photo you share specifically relates to Mecca. Saudi Arabia used to be very restrictive in allowing foreigners, muslim or not. In the past, I believe this mostly meant that a pilgrimage to Mecca (for non-Saudis) would only be possible when organised through officially recognised organisations, which typically would operate via your local mosque. So, your chances of being able to try and get into a 'Muslims only' area in Saudi Arabia, as a non-muslim would be so slim as to be practically irrelevant.
More recently, Saudi Arabia has started opening up access to the country to tourists, muslim or not. If you are a tourist and a muslim, and encounter a 'Muslim only' sign, I believe it very likely that, if questioned, your stating you're a muslim would be sufficient. That said, notwithstanding the language barrier, you might face some scrutiny. Though if you know what you're doing (show a particular respect, wash before entering a mosque, dress conservatively, etc.), I would suspect this questioning will be limited. If you're very western-looking, I would expect you'd primarily experience a lot of curiosity.
Not quite a full answer, perhaps, but it's also impossible to predict what would happen on every single occasion.
Addendum: Converting to islam does not require an external institution to 'prove' your conversion. However, as per this page, obtaining a Hajj visa requires a notarised certificate from an Islamic Center, for converts. Yet, with Saudi Arabia recently opening up the country to more regular tourists, I have no idea what that means in the context of visiting Mecca. Has the sign in the image of the OP been changed to 'Hajj visa holders only'?
Anecdotal addendum: I visited Al Aqsa (in Jerusalem) two days ago (April 2023), which currently is off-limits for 'tourists and non-muslims'. My clothes apparently gave me away as a tourist and, to enter, I was required to say the shahada in Arabic to enter.