addFieldToFilter() And Condition in magento2

Try with below code,

$multiselectupdate =$multiselect->getCollection()
                                ->addFieldToFilter('customergroup', ['eq' => $categoryname])
                                ->addFieldToFilter('customeremails',['eq' => $mail]);

Here is the solution for you to understand how Magento 2 use query operators to filter with addFieldToFilter

click

or simple in your case:

$multiselectupdate->addFieldToFilter('customergroup', ['eq' => "$categoryname"]);

Try following way:

$multiselect->getCollection()
->addFieldToFilter('customergroup', array('eq'=>$categoryname))
->addFieldToFilter('customeremails',array('eq'=>$mail));

No need to assign new variable again.

Debug:

echo $multiselect->count();