Lead conversion depends on more than just the name. It also depends on the sharing model, the user's permissions, and the configuration of Duplicate Management. See Convert Qualified Leads for the permissions you need to give your users.
Generally speaking, if the system won't let you select an account or contact, it's because there were no matching records, as determined by account name and last name (not email, phone, or address, just name fields) that the user can edit.
There's some leeway in what the system allows for name matches. For example, if the company name is Acme, and the account name is Acme, Inc, then Salesforce presents an option to convert into the existing account. Likewise, if the contact is Robert Smith, and the lead is Bob Smith, and they have the same account, Bob Smith can be converted to Robert Smith.
Generally speaking, if duplicates are a concern, consider enabling Duplicate Management. It will actually warn the user if they're about to create a duplicate, based on standard or custom matching rules to suit your organization's purposes.
If, after checking all your settings, you still feel like it's a system glitch, consider contacting support with some concrete examples. I've never seen the system not offer the correct records to convert to as an administrator, and for my users, I've always been able to trace it down to a sharing problem or a typo.