<form>
    <h1>Form 1</h1>
    <div><input type="text" name="field1" placeholder="field1"></div>
    <div><input type="text" name="field2" placeholder="field2"></div>
    <div id="RecaptchaField1"></div>
    <div><input type="submit"></div>
</form>

<form>
    <h1>Form 2</h1>
    <div><input type="text" name="field3" placeholder="field3"></div>
    <div><input type="text" name="field4" placeholder="field4"></div>
    <div id="RecaptchaField2"></div>
    <div><input type="submit"></div>
</form>

JavaScript part

<script type="text/javascript">
    var CaptchaCallback = function(){
        grecaptcha.render('RecaptchaField1', {'sitekey' : '6Lc_your_site_key'});
        grecaptcha.render('RecaptchaField2', {'sitekey' : '6Lc_your_site_key'});
    };
</script>

recaptcha script url

<script src="//www.google.com/recaptcha/api.js?onload=CaptchaCallback&render=explicit" async defer></script>