Checking my name from the PAN card database.

While filling up the data did you fill it correctly? You might have filled it wrongly. You mentioned that your name on PAN card and bank details are the same and no mistake is there. So I personally feel you might have done a mistake in filling. So why can't you verify your application and correct it?
When there is no mistake on your PAN card why should you see that? Otherwise, you can apply again by filling the correct name and other details for the GST registration. You have the PAN Card with you. The name on your Pan card will be there in their data also. So there you may not get any additional inputs.

However, you can verify your PAN details as per the following procedure.
1.Logon to 'e-Filing' Portal. www.incometaxindiaefiling.gov.in
2. Go to the 'Quick Links' Section. Click on 'Verify Your PAN details'
3. Enter the following details.
PAN Number,
Full Name (As per PAN),
Date of Birth
Choose the 'Status' as applicable
4. Enter the Captcha as in the image and click on 'Submit' to verify your PAN details.
If any mistake is there you will get the information there.

You can keep your PAN card, application form and bank pass book together and recheck on the spell errors if any. Check the writing styles of your initials also. Sometimes what we write may be mis spelt due to styles of writing. It is better to fill in capitals or lower case but by not using cursive letters.
If there is change in PAN card, it will take some time for rectification. Go through the rules and regulations of filling the GST application throughly before coming yo conclusions

Sometimes we mistakenly leave the blank space in the name which seems to be correct at the time of filling the form but later this creates a problem. You should recheck or reapply for the GST number through the portal. If you are sure that you have correctly filled the form then you should get your PAN card verified on the e-filing portal or NSDL website. There is no direct number for the PAN card verification, you can only get this verified on the portal.

You can get your PAN details verified by following the steps mentioned.
1. Login to NSDL or E-filling website
2. Fill the details, your Name, and Date of Birth and PAN number
3. Select the status of your PAN card - Individual, Company, etc.
4. Enter the captcha code an submit
5. You will get the status of your PAN card

If you find everything is correct then please contact to GST helpline number to get this checked at their end.
Goods & Services Tax - 1800 356 8633
You can also visit - https://selfservice.gstsystem.in/ to get GST level information.