Yes, you can have two logos for your brand.
You can have four, if you need. Don’t go crazy just yet. There are ways to do it right.
You should always have a black and white version and a color version of your logo. You can also have a square version and a more horizontal or vertical version for different applications. Now a days there are many places where you need to have your logo and sometimes the square size works best (apps, podcasts, avatars, social media etc). Then there are other places where your logo looks better not as a square (t-shirts, cars, business cards, banners) or perhaps that is your preferred version for your brand.
It’s OK to have two versions of your color logo and two version of your black and white logo if they are similar enough to never be confused as another brand. Your colors and fonts should stay the same. The general look should stay the same too. Perhaps it’s an adjustment on the sizing of elements to fit it into a square or cropping of certain elements to fit in the square.
When in doubt, hire a designer to help you with all of this. You want your logo to have integrity and look professional, always.
It’s always good to know what is your preferred logo and when it is OK to use the exception or the other version. These things can be outlined in your brand guidelines so that everyone (employees, bosses, vendors, printers, partners) that uses your logo knows how to use it.