Singing and playing starts with being able to already do one skill ,so if you can sing you practice playing, not playing and singing, and if you can play you practice singing, not singing and playing. Once you get the two developed they develop together, depending on how you use them, to be a natural skill you do without to much thought. It is an inter-changeable skill that once learned it is the quality of the material that will challenge you in future, not the act of learning any more but the application of the act. This will usually mean syncopation of the playing V syncopation of the voice rather than any tone or pitch of the voice.
The criteria differers on the way the strings are struck, picking note with the fingers is harder than strumming them with a pick.
Our natural instinct is to strum, if you lay a bass down and a small child comes to it they will strum it, not pick it, the natural instinct is to strum.
In strumming we have a motion to time 1-2-3-4 to if need be.
The tempo can be altered by the length of the strum, so we can find the beats within the length of the motion, or use the motion to define the beat.
With the fingers there is no motion as such to play to, but react to.
The reaction is the fingers, and depending on how many fingers used the re-actions becomes more complicated, unlike the strum which is one action so easy to control and time.
Those than can already sing rarely have as many issues learning to play bass as those that can play bass have learning to sing.
Those that can sing do so for many reasons, but mainly it is because they have a good voice they like the sound of, and are willing to use it.
I know many that have a good voice, but do not like the sound of it, so do not use it. With a guitar you can EQ your sound to one you do like, with your voice you hear what you hear.....like it or not, this is a different issue but a relevant one, you have to like the sound of your own voice.
In learning to sing and play you first need to be competent at one so you do it naturally without thinking, so if you do not know the lyrics, learn them, learn to sing them with no instrument and no lyric sheets.
As in playing, spend about 10minutes warming up your voice so it is ready to practice.
Using lyric sheets involves another layer of thinking, reading and singing and playing, so learn to sing the song un-aided, then learn to play it un-aided, this now reduces it to two things, how good the two are together is just practice. Write out the lyrics, see the changes, make marks or notes, practice the changes till it becomes natural.
When singing, sing out in a performance level volume, do not sing to your self, do not sing quiet, this is not the way it will be live, so sing out.
If that is embarrassing then you have doubts about letting others hear your voice, but the voice is a muscle the more it is used the more it develops, it needs to get warmed up to truly do you justice and give you the confidence to really use it.
For someone who can sing, learning to play is the same as someone who cannot sing, but with the skill of singing in place already whether it gets used does not reflect on how well the bass is learned.
As i said, singers can learn to play and sing easier than a player can learn to sing and play.
Part of this is we sing as children, we sing freely and without care as we grow up, and the opportunities are all around us to do so and the skill of singing is developed.......we do not have that with playing bass......bass playing comes many years after singing, or when we have stopped singing for whatever reason, usually when our voice breaks.
Some may go years without singing, some may even never sing for the rest of their lives....for whatever reason, but they just do not sing again because they do not like the sound of their singing voice.
Learning to sing to a bass is hard, you should learn to sing to a guitar or piano. Good bass players that can sing pitch their bass to their voice, not their voice to the bass, that is a big difference. So start easy, find songs where the rhythm of the bass is the same as the voice, get used to that, then find songs where the playing is against the voice, get used to that, then fill in all points in-between. Remember you may have to learn songs you do not care much for because they best show the skills you need to develop. But you are learning such songs for their educational value to your learning, how you use that skill once learned is up to you.
Listen to these songs for example, the each have a skill in them that a bass player may struggle to perform, but they appear in countless other songs, so see it as learning the skill, not the song
Take it Easy - The Eagles
His Latest Flame - Elvis Presley
Stand By Me - Ben E King
Sultans of Swing - Dire Straits
The good thing is stick with it as it gets easier as the skill develops, and do not be put off with the 'tone' of your voice as that changes a bit as you develop. In the end singing and playing is a skill that will benefit you, even if it is just harmonies, backing or reinforcing vocals, it all helps make you a complete player.