Users who purchase memberships will not see their memberships extended unless they have one of the default Dolphin memberships (i.e., Standard, Promotion, Non-Member), one of which is assigned by Dolphin when they create their profiles.
In other words, if a member would like to renew their paid membership, their membership will not be extended automatically until apparently, their current membership expired:
Ticket #2528: Payment Module - Automatic membership update after successful payment - PayPal
Note. The membership level will be added immediately after the payment was done if you have Standard membership level only. By default Dolphin has 'Promotion' membership level enabled. So, all newly joined members have Promotion membership level. If member already has some non-standard membership level then newly purchased membership level will be applied when the current one is expired.
Users should see their newly purchased membership below the current membership expires (see attached). How else will members & admins know the membership was renewed?