Jehovah's Witnesses believe a bit of both of those possibilities, depending on what kind of "honor" is being referred to. To the same degree as the Father is honored as "Judge of all the earth" (Gen. 18:25), they honor the Son as Judge. However, they do not honor the Son as being the Father or as God. Given the context of John 5:23, Jehovah's Witnesses qualify the "sameness of honor" assigned to the Father and Son to their respective roles as the judges of mankind.
Here are some Watchtower articles which discuss the meaning of John 5:23 (bold mine):
13 The Sender is greater than the one sent. Jesus himself said that. (John 13:16) God the Greater sent the Son the Lesser, for which reason Jesus also said: “The Father is greater than I am.” (John 14:28) The Father the Greater appointed the Son the Lesser to be a judge, committing all the judging with reference to mankind to the Son, who died sacrificially for them. If we humans respect the Father’s power to appoint, we ought to honor the one whom he appoints as judge. Just as we should honor God the Father who is “the Judge of all,” so we should honor the Son whom he appoints to judge mankind. Certainly if we do not honor the Son as God’s appointed judge, we do not honor God the Father who clothed his Son with judicial power. We cannot claim to honor God the Father and at the same time rightly ignore the Son as judge.
14 Our everlasting life depends on our thus honoring the Son as judge in the same way as we honor the heavenly “Father who sent him.” Today, by means of John’s written account in the Holy Bible, we are hearing Jesus’ words in this regard. If, after thus hearing, we do as Jesus said, namely, ‘believe on him that sent me,’ we shall have everlasting life. This life we shall enjoy in God’s promised righteous order of things under his kingdom.
ALL along the evidence has been mounting up from John’s own writings that Jesus Christ was the Son of God. This very fact in itself argues that Jesus as a Son was dependent upon God and was not equal to God. A son is not greater than his father, but must honor his father, according to God’s command. As God’s Son, Jesus said: “I honor my Father.” (John 8:49) How, then, can anyone say he was making himself God or the equal of God when he said: “The Father judgeth no man, but hath committed all judgment unto the Son: that all men should honour the Son, even as they honour the Father. He that honoureth not the Son honoureth not the Father which hath sent him”? (John 5:22, 23, AV) In those words Jesus was not telling us to honor him as being the Father or as being God. He did not say we were to honor the Son as much as the Father.
34 Look at Jesus’ words again and see why he said he was to be honored just as the Father is to be honored. Jesus said that the Father had appointed him to be judge, to act as the deputy or representative of God the Supreme Judge. Hence as God’s appointed Judge the Son deserved to be honored. By honoring the Son we show respect for God’s appointment of the Son as Judge. If we do not honor the Son as Judge, then we do not honor “the Father which hath sent him.” But that does not mean we honor the Son as being God himself or honor the Son as much as God himself, who sent the Son.