Why is a MAC verification algorithm required when we can simply create a new MAC and compare the received one with the one calculated? The process of verifying integrity through MAC involves two separate algorithms (as mentioned here). On the server side, a signing algorithm runs which calculates the MAC of the message using the shared key, while on the client side, a verification algorithm runs which uses the secret key, the received MAC and the message to verify whether the message was altered in transit or not.
But we can simply dispense with a separate verification algorithm by simply calculating the MAC of the received message using the shared key. If the calculated MAC and received MAC are the same, the message has not been altered.
Is that what actually happens in all MAC verification algorithms?
Signing and signature verification are concepts related to public key cryptography.
In case of message authentication codes, there is a single secret key and one function:
T = MAC ( K, M )

Where:

T - tag
MAC - MAC function
K - secret key
M - message

Messages can be signed by either the client or the server and verified by the other, depending on the case. Both parties calculate MAC the same way, using the same function and secret key, over the same message. There is no separate function for MAC verification. The only extra step on the receiving end is associated with comparing if both received and calculated tags match, and then acting accordingly. But this has nothing to do with a MAC algorithm itself.