There are (at least) two similarities.
One, both are from God.
In Corinth, Christians had some difficulty with buying or eating meat. It was very likely that the supermarket sold fine cuts that had initially been offered to idols. Could Christians eat this type of BBQ?
Yes! Idols are nothing, these ‘gods’ don’t exist. But the food is tasty and a good thing from God. So enjoy your steak sandwich. (Personally, I recommend sweet chili sauce or a decent relish, rather than the commercial red sugar syrup named ‘tomato sauce.’) See 1 Corinthians 8.
Likewise, apostles – Paul himself – were not bad, or a terrible burden. Paul did a good work, which was a work from God. Therefore he had every right to live by this. In other words, to be financially supported by those blessed by his apostleship. See 1 Corinthians 9:1-11.
Yet, secondly, both can be given up for love.
If a Christian of tender conscience found out that your steak sandwich included meat butchered in an idol’s temple, a strong Christian would naturally not eat it (1 Cor 8:13). The reason: nothing to do with the quality of the steak, everything to do with love for the weaker Christian. It’s never worth hurting the faith of a Christian, never worth leading a Christian to do what he/she thinks is ungodly. Not ever for steak!
Likewise Paul, after strongly insisting that he had the right to be supported as Christ’s apostle, says ‘I never sent you an invoice and never will.’ See 1 Cor 9:12 and following.
Paul refused to take the cash and insisted on working for free. He chose to serve, not insist on rights. He did so to better preach the gospel. The ‘better’ is important. Paul already was preaching the gospel of Jesus, but he strove to do it more clearly, more accurately, with less distraction.
Thus, Paul’s apostleship changed even the way he ate at a BBQ. And was the model Corinthian Christians were also to follow.
So too for us. While we love the good things of this world – knowing them as real gifts of God – we will always hold them very loosely, that we can choose to give them up. This light hold is so we can easily choose the better way to share the gospel, the wonderful news of Jesus as Lord.