Can Base64 be seen as a way of encryption if we use different index table? As we know, the index table (which maps bytes in [0, 63] to characters) of Base64 encoding is public, so everyone can decode Base64 strings from others.

But what if we use a shuffled index table and keep it secret? Can we see Base64 with this new table as some sort of safe encryption, and the shuffled table as cipher?
Can we see Base64 with this new table as some sort of [..] encryption [...]?


Yes, technically it does constitute a form of encryption as knowledge of a secret key is required to recover plaintext from ciphertext.


  Can we see Base64 with this new table as some sort of safe encryption [...]?


NO! This is essentially a monoalphabetic substitution cipher applied on the Base64 encoded data. This can be seen as you can always take your table $T$ and then find a new $T'$ that undoes the base64 table encoding first and then applied after standard Base64 encoding.

Now while the key size is (theoretically) respectable with $\log_2 (64!)\approx 296$ bit, this cipher suffers from all the usual problems such ciphers have, including:


Ciphertext-only attacks are possible using frequency analysis, as there are only 64 valid plaintext values and they will probably occur with different frequencies.
Known- and chosen plaintext attacks are devastating as there are only 64 mappings to be uncovered and usually plaintexts have quite a bit of structure allowing to guess the rest / uncover significant parts of the plaintext.