If hypothetically Ethereum market capitalization reach Bitcoin market capitalization, will Ethereum transaction fees be equal to Bitcoin transaction fees? Or what would be cheaper?
Probably cheaper, but it's complicated.
Fees are not directly related to market capitalization. There may be an indirect relationship, in that including extra transactions makes it more likely that blocks will be orphaned, and the loss that results from a block being orphaned is denominated in ETH, so it increases when the value of 1 ETH goes up, and miners may expect to be compensated for this greater loss rate in higher fees.
However, the main driver of high fees in Bitcoin as of June, 2017 is that there are more transactions requested for inclusion in blocks than the blocks can accommodate. This is in turn happening because Bitcoin has a hard-coded cap on the amount of capacity that can be handled by one block. (This is based on the size of the transaction data, and is set at 1 MB.) There are various proposals to change bitcoin to raise this cap or accommodate more transactions inside it, but none have yet been activated on the live network. When there is too little capacity to meet demand, users raise the fees they offer to attempt to make their transactions more attractive to miners than other users, and the resulting bidding war can make fees quite high.
Ethereum also has a capacity limit, called the Block Gas Limit. Currently Ethereum transactions are not usually hitting this limit, so fees are mostly substantially lower than Bitcoin's fees. However, there have been occasional spikes caused by crowd-sales where fees became temporarily very high.
Unlike in Bitcoin, the Ethereum Block Gas Limit can be raised dynamically by miners voting for an increase. As long as they increase the limit to stay ahead of demand, it seems unlikely that fees will rise particularly high. However, it is possible that either miners will choose not to raise the limit, or technical scaling limitations will make it impractical to raise it any further. At that point Ethereum may see very high fees, as we have seen in Bitcoin.