I have used DocuSign in the context of contracts with a managed service provider for an end client. I don't like it and it was sprung on me at a contract renewal.
It seems to be a bit odd to sign things with a signature that looks nothing like your own real signature - it gives several signatures to choose from and I think it is supposed to remember the one picked but it never did for me so I could change it for each document.
A friend that works in corporate contracts says signing with DocuSign is legally binding in the UK.
I thought it was a bit annoying to have to do it that way as it was convenient to show the managed service provider the real thing in person, but the policy was use DocuSign to receive/return signed contract or no contract.
One of my concerns is if you were not expecting a document to sign, and the sender of the DocuSign document got your email address slightly wrong (or you mis-typed it into some form somewhere, or their database got corrupted or hacked) and by chance that was a live email address of a third party, that third party could be naughty and sign the document. Not sure how you would get out of that situation as that third party had effectively signed a contract on your behalf without your consent/authority.
Similarly if someone hacks into your email account or you inadvertently open email that has hidden malware, they can access all the stuff you have on DocuSign secure servers.
Consider using different email accounts for different purposes.
It sounds like the owner of that apartment has had too many problems with people booking under false details or using stolen credit cards and not paying and been untraceable.
I have used booking dot com to book hotels over the years and never needed to provide proof of ID or address.
Trust works both ways. There is nothing to stop the apartment owner/agent from saving a copy of your DocuSign document and selling it to passport forgers. The same could be said for hotels that take a copy of passport pages at hotel check-in.