I am designing an application where I have two brands' logos to be displayed on the header. Positioning two brand logos next to each other looks odd.
Which is the best way to position the logos on the header?
This is very difficult to answer because I don't know what your logos look like.
However, I would consider putting the Brand Logo 2 on the left side before the user and notification icons. This is because on many websites these things will be the very last things in the header and having them just before the logo. (Like in your second picture) could be confusing.
This might end up looking horrible though (depending on the icon) so you should definitely have a bunch of end users look at multiple possibilities to chose.
Would it be possible to have the customer's logo up top (since they're the paying customer) and your logo in the footer, in a "powered by" sort of display? Similar to a customized WordPress site. The client customizes the site with their brand, but the very bottom prominently states that the site was created with WordPress. Just a thought. It's hard to know without more context of the site, the industry, the logos, etc.