There's more to it.

With on-chain transactions, it is block space that is scarce. So your fee with an on-chain transaction is based on how much space your transaction takes (priced in sats per vByte), not the value being transacted.

With Lightning network, it is channel liquidity that is scarce. So your fee with an LN transaction is based on the channel liquidity available by node operators at the fee your are willing to pay. Thus as fees go, the higher the amount that you are transferring the the amount of fee you will need to pay will be higher as well.

If I paid 37 sats to send 5,000 sats via LN, I'ld be disappointed. If I paid 37 sats to send 1,000,000 sats I'ld have gotten a fairly decent deal.

Here's more on fees: