An example, two young boys on the playground:
Sam: “Wow Tom, your pretty tall.”
Tom: “I am tall, but my dad is taller.”
Sam: “yeah, but my dad is even taller than your dad.”
It can be used for emphasis. However, the examples you mentioned can be said either way. You can say “I am tall, but my dad is taller” or “I am tall, but my dads even taller”. The sentence alone has relatively the exact same meaning. Even, in my opinion, it is used more often when the speaker is trying to emphasize what they are comparing as being better or more exciting or more impressive than the other.
Another example:
“I am a great baseball player, but I am an even better soccer player”
I already said I am great at baseball, so to get the message across that I play soccer better than I play baseball I can use ”even”. Of course you can always say “I am great at baseball, but I am better at soccer”. But if you wanted to emphasis that it’s impressive that you could be better than you already are you could use “even better”.
I hope this helps you a little bit, English can be confusing sometimes.