Yes, you can. If you use Google CSS, you can verify multiple accounts with your domain name, but you can only claim one account on your domain name. This means that with Google CSS you can only use one Merchant Center actively to serve ads. With any other CSS, including Bigshopper CSS, it works differently. Every domain name can be verified and claimed by a Google CSS Partner at any time. In fact, it is possible to claim multiple Google Merchant Center accounts as a CSS Partner for one domain. This is also necessary in a number of cases. Suppose you have a .com domain name with 3 different languages (e.g. the Netherlands, German and French). Then under Bigshopper CSS you need to use 3 accounts for the same domain name: one Merchant Center account for the Netherlands, one for Germany and one for France. Google requires this, because there is a link to "By Bigshopper" under the Shopping ads, if a user in Google clicks on it, this user will go to the site of Bigshopper (and see related articles there). Google requires that the user is shown the related products in the right language of the right country. Therefore, each country must be linked to Bigshopper CSS in that specific country.
Please note that if you use affiliate marketing, it is also possible for affiliates in your program to verify and claim your domain name. If you do not wish to do so, please indicate this in the terms and conditions of your affiliate program. Bigshopper does not advertise affiliate feeds in the Merchant Center in any country or in any situation. Other CSS Partners will do this, please note this. If you work with Google CSS, you can see which CSS Partner has claimed your domain and possibly advertises with your products (whether or not as an affiliate). If you want to check this, you can log in to your Merchant Center. Then choose "Shopping ads" from the menu on the left and then select Configuration. You will now see a dashboard with all the CSSs that have claimed your domain name and may be advertising your products. If you use a CSS, for example Bigshopper CSS, you will no longer see this dashboard in your Google Merchant Center. You can then create a new Google Merchant Center for free (via Google CSS), where you verify and claim your domain name. If you do this, you will not lose your claim from your Merchant Center that is covered by Bigshopper CSS (despite the fact that you can receive a warning). After making a claim, you will be able to see in the Google CSS Merchant Center (which does not need to be active) the dashboard with all the CSS Partners who may be advertising your products.