Is a tweakable block cipher still considered deterministic in nature? "a block cipher is a deterministic algorithm operating on fixed-length groups of bits, called blocks, with an unvarying transformation that is specified by a symmetric key. - Wikipedia " 

But with a native tweak support inside a block cipher, one can achieve "varying" transformation with same symmetric key but different tweak. i.e probabilistic encryption without changing the key or using the block cipher in some mode .

Now is a block cipher with native tweak support still considered a deterministic encryption ? The only determinism being the length of the output (that is same as length of the input).
When talking about encryption with block ciphers, you should distinguish block cipher itself from the mode of operation that employs the block cipher. 

Block cipher takes an input $P$ of fixed length $n$ and transforms it under key $K$ (and possibly under other parameters such as tweak $T$) to output $C$ of the same length $n$:
$$
E:\; P \xrightarrow{K(,T)} C.
$$
It is always deterministic in the sense that $C$ is entirely determined by the listed parameters.

A mode of operation $\Pi$ is used to turn a fixed-length block cipher into a variable-length encryption scheme. To achieve semantic security, modes of operation typically uses deterministic nonces or random initialization vectors. In the latter case the resulting schemes are called probabilistic. 

The use of tweaks is somewhat orthogonal to this classification. Both deterministic and probabilistic modes may use tweaks to further parametrize the encryption (e.g., in disk encryption).