Your manager is right, I think. You can submit a self-assessment tax return at the end of the year (it's technically due on 31 January 2022 if you are submitting online), and your current employer would not be notified.
Regarding the self assessment: the side income from your second job is taxed at the same rate as the income from your full-time job (20%, based on what you have described). However, you can reduce the amount of taxable income because HMRC gives you a £1,000 trading allowance on any side income you earn.
So for example, if you earn £135 / month * 12 = £1,620 over the 2021/22 tax year, your taxable income would only be £1,620 - £1,000 = £620. Your tax bill would then be £620 * 20% = £124.
Filling out a tax return and doing all the forms can get pretty annoying (I'm currently a freelancer myself). You might want to look into some online services that automate the entire process for you, e.g. something like earnr (if you're just starting off) or Xero, if you're looking for something more comprehensive.