No, not exactly, and not even close except for a special case or 2
A bit is an abstract unit of information.
A transistor is a physical thing, a current (BJT) or voltage-controlled (FET) switch, more or less. Modern digital logic uses FETs (CMOS), so voltage-controlled.
In DRAM, a transistor and a capacitor can temporarily store a bit (when used as part of a DRAM array with other transistors for addressing and so on).
Or a single transistor plus a resistor can (in RTL logic) implement a NOT gate which flips 1 bit. (A CMOS NOT gate takes 2 transistors.)
In general no, it's not sensible to say that a transistor is a bit. If you're talking about how many transistors are needed to implement a cache or register file with a given capacity, then it becomes relevant that an SRAM cell usually consists of 6 or more transistors. (https://en.wikipedia.org/wiki/Static_random-access_memory#Design). Using more than 6 can help it work more stably at low voltages, saving power. (Modern CPUs have such tiny transistors that they're more limited by power density than number of transistors. See http://www.lighterra.com/papers/modernmicroprocessors/ - it's a very good overview of CPU-architecture stuff, but at a higher level than individual transistors.)
When not part of a whole array of memory, a bit might be stored by a flip-flop / latch made of multiple transistors to implement clock, set/reset, and input, depending on what type of flip-flop.
That wikipedia link shows some flip-flop designs. A bit is the state of voltages on those transistors, not the transistors themselves. Or you can say that the flip-flop has a storage capacity of 1 bit.