There are not very many other members of this group where I live, and I'm often happy to meet others so we can talk about relevant stuff that most of my other friends don't really 'get'.
Typically minorities form local community groups that allow for meet-ups and social connection. Have you tried looking for these and contacting/meeting with them ?
This would strike me as the most sensible approach to your problem.
As a student I would often just ask someone, if I had an inkling based on their name or something they'd said, "Are you ish?". Of the maybe ten times I asked this, the answer was always 'no', and it became a bit awkward and I would blurt out "Oh sorry, I am, I just ask everyone that" (partly so they wouldn't think I was asking for anti-ic reasons).
Keep in mind that student life is not the more formal and legalistic setting of work. Many organizations have rules against discrimination which effectively make inquiring about these personal things potentially at odds with protocol and even contract conditions.
I've recently started working at a medium-sized company, and I've just realized there's a guy here with a quintessentially -ish name.
Work is, once more, an inappropriate venue for what is de-facto you pursuing your own social agenda based on ethnicity and religion. I advise against this.
Even if this person has a background as you describe/hope they may also be strongly opposed to being reminded of this. They may have left this behind them and actually find it annoying to be reminded of it. Great caution is needed with these issues.
I'd love to find out if he's a part of the group, for work and other purposes
Work is not for other purposes. Work is for earning a living. Focus on that in work. Do not jeopardize that for social goals.
(e.g. asking how HR/our bosses responded if he asked for time off on religious holidays
HR or your boss could tell you that more authoritatively. :-)
, potentially going to services together,
That's a social, perhaps even a very personal issue which may be quite inappropriate to pursue in the context of a workplace.
seeing if he can link me up with some other local s, exchange holiday greetings, potentially chat in a relevant language, etc).
That's something you can achieve by seeking a local community group (and there probably is one). Online is another option for this.
But mixing work, religion and social is asking for problems.
But I don't want to make things weird and awkward with a colleague. What's a good way to approach this? Or should I just mind my own beeswax?
That last option sounds right on the money to me. :-)
Your ethnic and religious "persona" are clearly important aspects of your life to you, but you must avoid overlapping this with work.
The most typical way to "flag" your own ethnicity and religion is to wear something discrete, like a necklace or similar small item that displays your interest. For example, I'm from a Catholic background and typically someone might wear a small cross on a necklace or wrist chain. A former colleague who became a preacher had a (discrete) cross on his desk, but was otherwise not in any way pushy about his beliefs.
Check what's acceptable with HR/your boss. There are rules, often a result of legislative requirements, which you need to respect. Work within those.
Give some thought to keeping work and personal apart.
Discretion in the workplace is generally advised in these matters.
Search for social groups in the community at large that will offer an outlet to your need to be around people from a similar ethnic and religious background more often than currently. This is a perfectly reasonable need, just not one appropriate to the workplace.