I believe you are misinformed. InDesign doesn't edit websites. It has a weak ability to create HTML from InDesign layouts, but that's all.
Perhaps someone used Adobe Muse (which visually looks like InDesign in some ways).
Adobe programs to edit web sites are Adobe Muse or Adobe Dreamweaver.
Some other applications of the Technical Communication Suite can also produce web sits, but more on focus on online manuals help menus as you find it in all Adobe Applications in hte Help menu in combination with FrameMaker which is the standard for manuals.
As mentioned, don't use InDesign, this application is for producing printed and electronic publications.
Also, although Muse is an excellent application for producing websites, the site may have limited functionality and you need to work within a Muse workflow.
Dreamweaver, which is available as part of Adobe CC, would be ideal for your purpose. It includes the facility of downloading a site, editing it and uploading it. But it's not easy to understand, so you'll probably need some training. Lynda.com have some excellent online tutorials, such as: Dreamweaver CC Essential Training | Lynda.com
(I have no connection with Lynda.com!)
Well, whatever software you eventually use to edit, you are going to need to know how to download it. Because you don't edit sites "in situ", you download them, work locally, and upload as needed.
Look for a tool specifically to capture/download web sites. Though if you have FTP access it may be better because there could be hidden files.
It may be worth mentioning Creative Cloud, in case this is not realised. A Creative Cloud subscription gives InDesign and Illustrator and Photoshop; and Muse and DreamWeaver and a lot more. Some of these are best for print, some for web, some cross over. But the reason you may have been told that InDesign was used is that it might be that the creator had a Creative Cloud subscription, maybe bought to use InDesign, and then used other bits of the suite for web design.
Tip: don't look for a single app which does everything. With Creative Suite most tasks use a mixture of apps.
Muse does allow for editable regions through a browser, but that still doesn't explain thinking the site was done with InDesign.
Thank you for the input! That helps alot!