For many options and futures I can see that
Contract Size = Tick Value / Tick Size
Are these values always related like this, and if so what does the relationship mean?
If we take 3m CME ED Libor IR futures, the value is \$2,500 * IMM - that is, the movements in the contract price are amplified by \$2,500. If the contract moves from 95.00 to 95.01, that means a movement in the value of a position of 1 contract by +0.01*\$2,500=\$25. That relationship does not change, but the tick size does change when each ED future becomes the front contract. At that point the minimum tick changes from 0.005 (\$12.50) to 0.0025 (\$6.25). The factor is the same, so a movement by 0.01 is always $25, but now I can trade at 95.8875 instead of only at 95.8900 or 95.8850.
So really the tick value is derived from the notional contract value and the tick size, not the other way around.
By contrast, ICE states that a 1m USD SOFR future has a contract value of \$12,000,000, and has a tick size and value of 0.0025 and \$25 respectively. But 25/0.0025 does not equal \$12m. Instead they are quoting the notional on a 1y rate, as the SOFR rate is quoted annually, and applying a factor of 1/12 to get the value of the change for a 1m future.
This matches with a 3m ICE SOFR future whose contract value is \$4,000,000, where the factor (for 3m instead of 1m) is 1/4, giving the same \$1,000,000 multiple between tick size and value; they are also \$25 per 0.0025 tick.