Here is the phrase which I am reading:
The only fully equipped centre of its kind, TANUVAS’ critical care unit has come to the rescue of many a pet
Source: A go-to place to save animals in their golden hour
Here is the phrase which I am reading:
The only fully equipped centre of its kind, TANUVAS’ critical care unit has come to the rescue of many a pet
Source: A go-to place to save animals in their golden hour
Kind is a noun that is synonym of sort.
A group with similar characteristics, or a particular type. (Cambridge Dictionary online).
The sentence means that there is no other centre as well equipped as TANUVAS’ critical care unit. As it says further down in the article, it's "the best".
Of many a means "a lot of". "Many a" is archaic, you would not use it in everyday conversation.
Of its kind
This means, in this case, 'which can provide the service that it does'.
In general it means 'which has its particular or similar characteristics, attritubutes or facilities.
https://en.oxforddictionaries.com/definition/of_its_kind
Of many a pet
This means 'of many pets'.