And call no man your father upon earth (Mathew 23:9)

Matthew 23:9 And call no man your father upon the earth: for one is your Father, which is in heaven.

Does this mean that you should not call you dad, father? As that is what I think it implies, or is it that the context of the word father is different in the verse?

Thank you for the question. Before starting the answer, I would like to say that Bible is the only Book, that gives us answers to this day and for future too. Here is your answer:

And call no man your Father... This does not, of course, forbid us to apply the term to our real father. Jesus here, is talking to the people about the scribes and pharisees. They were people who followed the law, but didn't really love the law. They did it, not for the love of God, but to be recognized by people. Those pharisees and scribes worked to attain a holy title, but did nothing truly holy to get it. (Think faith without works is dead... but in reverse)

The word “father” also denotes “authority, eminence, superiority, a right to command, and a claim to particular respect.” In this sense it is used here. In this sense it belongs eminently to God, and it is not right to give it to people. Christian brethren are equal. Only God has supreme authority. He only has a right to give laws; to declare doctrines that shall bind the conscience; to punish disobedience. The Jewish teachers affected that title because they seem to have supposed that a teacher formed the man, or gave him real life, and sought, therefore, to be called father. Christ taught them that the source of all life and truth was God, and they ought not to seek or receive a title which properly belongs to him.

Jesus meant the full sense of this noble word for our heavenly Father. “Abba was not commonly a mode of address to a living person, but a title of honor for Rabbis and great men of the past” (McNeile). In Gethsemane Jesus said: “Abba, Father” (Mar_14:36). Certainly the ascription of “Father” to pope and priest seems out of harmony with what Jesus here says. He should not be understood to be condemning the title to one’s real earthly father. Jesus often leaves the exceptions to be supplied.

Conclusion
Father (as in daddy) is fine. Father (as in priest) is wrong.