In my experience there are a couple things to check. Template overrides in layout files may be incorrect, the easiest way to check if this is the case is to delete your custom layout file for that page, rebuild and check the form reCaptcha again. If this fixes reCaptcha on your page, look for examples like the following in your custom layout file. In yourmodule/etc/customer_account_forgotpassword.xml for example, if you are using:
....
<block class="Magento\Customer\Block\Account\Forgotpassword" name="forgotPassword" template="MyCompany_Module::form/forgotpassword.phtml">
.....
This would be incorrect and break reCaptcha because the 'forgotPassword' name reference already exists in Magento. You have to use something like this instead:
....
<referenceBlock name="forgotPassword">
<action method="setTemplate">
<argument name='template' xsi:type='string'>MyCompany_Module::form/forgotpassword.phtml</argument>
</action>
</referenceBlock>
....
If you are still finding issues even with the layout file for the page deleted, it may be a CSP issue. Try adding the following as csp_whitelist.xml to yourmodule/etc/ folder:
<?xml version="1.0" encoding="UTF-8"?>
<csp_whitelist xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:noNamespaceSchemaLocation="urn:magento:module:Magento_Csp:etc/csp_whitelist.xsd">
<policies>
<policy id="script-src">
<values>
<value id="google-recaptcha" type="host">https://www.google.com/recaptcha/</value>
<value id="gstatic" type="host">https://www.gstatic.com/</value>
</values>
</policy>
<policy id="frame-src">
<values>
<value id="google" type="host">https://www.google.com/</value>
</values>
</policy>
<policy id="img-src">
<values>
<value id="doubleclick" type="host">https://stats.g.doubleclick.net/</value>
</values>
</policy>
</policies>
</csp_whitelist>
