This paper proposes a textless training method for many-to-many multilingual speech-to-speech translation that can also benefit the transfer of pre-trained knowledge to text-based systems, text-to-speech synthesis and text-to-speech translation. To this end, we represent multilingual speech with speech units that are the discretized representations of speech features derived from a self-supervised speech model. By treating the speech units as pseudo-text, we can focus on the linguistic content of the speech, which can be easily associated with both speech and text modalities at the phonetic level information. By setting both the inputs and outputs of our learning problem as speech units, we propose to train an encoder-decoder model in a many-to-many spoken language translation setting, namely Unit-to-Unit Translation (UTUT). Specifically, the encoder is conditioned on the source language token to correctly understand the input spoken language, while the decoder is conditioned on the target language token to generate the translated speech in the target language. Therefore, during the training, the model can build the knowledge of how languages are comprehended and how to relate them to different languages. Since speech units can be easily associated from both audio and text by quantization and phonemization respectively, the trained model can easily transferred to text-related tasks, even if it is trained in a textless manner. We demonstrate that the proposed UTUT model can be effectively utilized not only for Speech-to-Speech Translation (S2ST) but also for multilingual Text-to-Speech Synthesis (T2S) and Text-to-Speech Translation (T2ST), requiring only minimal fine-tuning steps on text inputs. By conducting comprehensive experiments encompassing various languages, we validate the efficacy of the proposed method across diverse multilingual tasks.