This statistical figures from CoinMarketCap show clearly that it's Bitcoin and Others. There is no huge rivalry here, and never will it be.

Market dominance based on market cap is an overrated thing.
The market capitalisation of any asset can never be overrated, as it is the direct reflection of the status and economy of that asset. It now depends on what you are using it to measure. For instance, in Bitcoin dominance (crypto space), though it will still not be overrated regardless of the application, yet it may not be appreciative in percentage if more assets are involved. But this doesn't affect the progressiveness of the asset in most cases. Bitcoin is a good example, it continues to rise even as the dominance percentage may not necessarily increase due to too many coins/tokens involved. Thanks to the market capitalisation, it can't be stated as overrated.
If a coin like Shiba Inu, for example, gets to $1 per coin (I know it's almost impossible because of how shit it is), that will make its market cap around $589 trillion, which will mean it has a higher market cap than bitcoin. Does that make it more dominant?
This looks almost impossible. Notwithstanding, if it happened, be assured that the market cap of Bitcoin will be measurable in multiples of that of Shib at the time. So Bitcoin dominance will always prevail regardless of the progress made by other coins/tokens.