Do I need a wildcard cert
You would need a wildcard certificate if you planned on using multiple records with a single certificate. Typically, a non wildcard cert will be signed with the naked domain and one other name you choose. Default is usually "www". Each CA is a little different in the manner they present these options.
If you are planning on using that certificate with "www" and the name you have mentioned in the comments, then you would indeed need a wildcard cert.
If you are only using that one name or the naked domain with the cert you purchased and the cert contains that name, then perhaps they are doing something incorrect on their end, or there is a miscommunication occuring. If either party is lacking coffee, this can happen.
In your case, I can not clearly answer this just yet, as the cert being presented on the name you mentioned is for the ISP. Perhaps they are using SNI in which case the default cert could be for their site and this would be expected.
This Thread offers great insight into this question as well. You may also want to read up on how to validate your SSL certificate using OpenSSL so that you can be sure what you purchased contains the name of your site.