If you're going to be regularly working on each other's code, you're going to need to be able to read it. Other people might even need to read it.
You can learn to read different code styles, but the problem emerges when you have to switch mindsets when you're looking at different sections of code. There's a very noticeable cost involved in this switch. Sometimes it's unavoidable, like switching from reading a Functional lang to reading Object Oriented code, but when you have to switch how you read just to read random pages of your site's code, it adds up fast.
If you both need to work on each other's code regularly, that cost is going to add up, and you're probably going to get into some discussions regarding code style: "Why the %$^% are you doing that?"
Additionally it is important that even if all your HTML pages are separate and don't need to be consistent in how the use IDs, classes ect, your CSS and your backend do need consistency. For your CSS to make sense you need to work out how you're going to use classes vs IDs and your naming for elements. Your forms are going to need some standards because the variable names you use for your forms go back into the server layer, and they need to map well to database inputs where relevant.
I also certainly wouldn't discard the notion of readable code since it's a markup language rather than a programming language; if anything it matters more in markup because your code isn't compiled, and non-semantic markup might actually display improperly or have accessibility issues.